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THE 
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*3/flONOMY  LIBRARY 


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ASTRONOMY  LIBRARY 


> 


LIBR  ARY 

OF  THE 

ASTRONOMICAL  SOCIETY 
OF  THE  PACIFIC 


AN 


OTRODUCTIOI 


TO 


PRACTICAL  ASTRONOMY, 


A    COLLECTION    OF 


ASTRONOMICAL  TABLES. 


BY 


ELIAS  LOOMIS,  LLD,, 

PROFESSOR    OF   MATHEMATICS    AND    NATURAL   PHILOSOPHY    IN    THE    UNIVERSITY 

OF   THE    CITY   OF    NEW   YORK,    AUTHOR   OF   A    "COURSE   OF 

MATHEMATICS,"    ETC. 


NEW    YORK: 

HARPER  &  BROTHERS,  PUBLISHERS, 

329     &     331     PEARL    STREET. 
FRANKLIN     SQUARE. 

1855. 


Entered,  according  to  Act  of  Congress,  in  the  year  1855,  by 

HARPER  &  BROTHER^ 
In  the  Clerk's  Office  of  the  Southern  District  of  New  York. 


L  i 


ASTRONOMY 

fc  LIBRARY 


PREFACE, 


THE  rapid  advance  in  the  cultivation  of  Practical  Astronomy 
which  has  recently  been  made  in  the  United  States  is  one  of 
the  most  encouraging  features  of  the  age.  It  is  less  than 
twenty-five  years  since  the  first  refracting  telescope,  exceeding 
those  of  a  portable  size,  was  imported  into  the  United  States, 
and  the  introduction  of  meridional  instruments  of  the  large  class 
is  of  still  more  recent  date.  We  may  now  boast  of  two  Observ- 
atories, liberally  equipped  with  instruments  of  the  best  class, 
and  provided  with  a  permanent  corps  of  observers,  as  also  a 
considerable  number  of  other  establishments  more  or  less  com- 
plete, and  a  still  larger  number  of  telescopes  of  dimensions  ade- 
quate to  be  employed  in  original  research. 

This  large  increase  of  instrumental  means  of  research  has  not 
only  been  attended  by  a  corresponding  increase  of  practical  ob- 
servers, but  also  by  an  increase  of  astronomers,  who  are  able  to 
apply  their  observations  toward  the  testing  and  perfecting  of 
astronomical  theories.  Not  only  have  the  latitude  and  longi- 
tude of  numerous  places  in  the  United  States  been  accurately 
determined,  but  a  large  number  of  fixed  stars  have  been  care- 
fully observed  and  catalogued  ;  improved  methods  of  observation 
have  been  invented  ;  the  places  of  the  different  members  of  our 
solar  system  have  been  accurately  observed  and  compared  with 
the  best  tables  ;  new  tables  have  been  constructed,  claiming  an 
accuracy  superior  to  any  thing  heretofore  known  in  Europe  ; 
and  we  have,  at  last,  our  own  nautical  ephemeris,  which,  it  is 
hoped,  will  contribute  to  hasten  the  era  of  our  national  scientific 
independence. 

While  the  attention  of  so  many  persons  is  thus  earnestly  di- 
rected to  the  improvement  of  Practical  Astronomy,  the  want  of 
a  suitable  text-book  on  this  subject  has  been  extensively  felt. 
Some  work  has  been  needed  which  should  not  only  give  an  ade- 
quate description  of  the  instruments  required  in  the  outfit  of  an 


MS77IOO 


iv  PREFACE. 

Observatory,  but  which  should  also  explain  the  methods  of  em- 
ploying them,  and  the  computations  growing  out  of  their  use. 
No  work  of  this  description  has  hitherto  been  attempted  in  this 
country,  if  we  except  one  or  two  treatises  whose  scope  was  con- 
fessedly far  too  limited ;  nor,  so  far  as  I  am  aware,  does  there 
exist  in  the  English  language  any  work  which  meets  the  de- 
mand in  our  country.  Pearson's  Practical  Astronomy  was  un- 
dertaken with  a  somewhat  similar  object  in  view ;  but  this  is  a 
work  of  inconvenient  bulk,  of  heavy  expense,  and,  withal,  fur- 
nishes the  student  with  very  little  insight  into  the  methods  of 
computation  now  most  generally  adopted  by  astronomers ;  nor 
have  I  met  with  any  work  in  any  foreign  language  which  ap- 
peared to  me  exactly  to  meet  the  wants  of  our  own  country. 

The  following  are  among  the  different  classes  of  persons  for 
whom,  it  is  believed,  a  work  on  Practical  Astronomy  was  needed : 

1.  Amateur  observers,  who  have  in  their  possession  astronom- 
ical instruments  which  they  wish  to  employ  to  the  best  advant- 
age, and  feel  the  need  of  more  specific  instructions  than  can  be 
gathered  from  the  elementary  text-books  on  Astronomy. 

2.  Practical  surveyors,  engineers  employed  on  boundary  and 
government  surveys,  astronomers  employed  in  determining  the 
situation  of  light-houses   and   other  important  points  on  the 
coast,  the  conductors  of  expeditions  of  discovery,  whether  by 
land  or  sea.     Indeed,  every  person  who  has  occasion  to  engage 
in  astronomical  computations  feels  the  importance  of  having 
before  him  a  volume  which  furnishes  the  formula)  for  his  use, 
and  tables  to  facilitate  his  labors. 

3.  There  is  a  far  more  numerous  class  of  persons  to  whom,  it 
is  believed,  a  work  on  Practical  Astronomy  may  be  highly  use- 
ful, viz.,  the  entire  corps  of  young  men  who  are  engaged  in  a 
course  of  liberal  education.     It  is  thought  that  the  study  of 
Practical  Astronomy  ought  to  be  incorporated  into  the  regular 
course  of  instruction  in  all  our  colleges  and  universities.     It 
may  be  said  that  very  few  young  men  in  our  country  ever  in- 
tend to  devote  their  time  to  the  business  of  astronomical  obser- 
vations ;  so,  also,  very  few  intend  to  become  practical  surveyors, 
or  navigators,  or  opticians ;  yet  we  include  surveying,  naviga- 
tion, and  optics  in  our  course  of  liberal  study,  prescribed  for  all 
indiscriminately,  whatever  may  be  their  ultimate  destination. 


PREFACE.  v 

Practical  Astronomy  has  claims  upon  our  attention  equal  to 
those  of  either  of  the  preceding  sciences,  whether  we  regard  it 
as  a  means  of  mental  discipline,  or  in  its  bearings  upon  other 
branches  of  study.  An  acquaintance  with  the  grand  principles 
of  astronomy  has  from  time  immemorial  been  regarded  as  an 
essential  part  of  a  finished  education ;  but  no  one  can  feel  a 
rational  confidence  in  the  results  announced  by  astronomers 
without  some  distinct  notion  of  the  methods  by  which  these  re- 
sults are  attained.  .When  the  student  is  told  that  the  sun  is 
ninety-five  millions  of  miles  distant  from  us,  and  that  light  re- 
quires several  years  to  reach  us  from  the  nearest  fixed  star,  he 
may  receive  these  doctrines  without  dispute  on  the  basis  of  au- 
thority, but  he  can  feel  no  adequate  conviction  of  their  truth 
without  a  knowledge  of  the  instruments  with  which  the  requi- 
site observations  are  made,  as  well  as  the  principles  upon  wrhich 
the  computations  are  conducted. 

It  is  believed,  therefore,  that  Practical  Astronomy  is  destined 
to  occupy  a  more  prominent  place  in  our  institutions  of  educa- 
tion than  it  now  holds,  and  it  is  hoped  that  the  present  volume 
may  contribute  something  to  so  desirable  a  result. 

The  preparation  of  this  treatise  has  been  attended  with  seri- 
ous labor.  No  considerable  portion  of  it  has  been  exclusively 
derived  from  any  single  work.  I  have  sought  for  materials  from 
every  source  within  my  reach — not  only  from  the  standard  au- 
thorities upon  this  subject,  but  also  from  Astronomical  Journals 
and  the  Annals  of  Observatories.  The  works  which  I  have  most 
frequently  consulted  with  success  are,  Pearson's  Practical  As- 
tronomy, and  Baily's  Astronomical  Tables  ;  Delarnbre's  Astro- 
nomic, and  Francoeur's  Astronomic  Pratique  ;  Briinnow's  Spha- 
rischen  Astronomic ;  Sawitsch's  Practischen  Astronomie,  and 
Bessel's  Astronomische  Untersuchun'gen. 

The  Tables  which  accompany  this  volume  have  cost  me  con- 
siderable labor.  Table  XVI.  is  entirely  original.  Doubtless  sim- 
ilar tables  have  been  heretofore  computed,  but  I  have  been  una- 
ble to  find  such  an  one  in  any  of  the  works  to  which  I  have  had 
access.  Several  of  the  tables  have  been  computed  entirely  ane«vv, 
although  similar  tables  are  to  be  found  in  other  works.  Of  this 
description  are  Nos.  XVII.,  XVIIL,  XXII.,  and  XXVII.  Others 
have  been  partially  recomputed,  extended,  and  modified  to  suit 


vi  PREFACE. 

the  size  of  the  page  or  the  plan  of  this  work.  Of  this  description 
are  Nos.  IX.,  XII.,  XIII.,  XIV.,  XV.,  XIX.,  XX.,  XXL,  XXIIL, 
XXX.,  and  XXXV.,  while  a  considerable  portion  of  the  remainder 
have  been  more  or  less  modified  in  form  or  substance.  There  is 
not  a  line  in  the  entire  volume  which  was  not  sent  to  the  printer 
in  manuscript,  and  large  portions  of  the  work  have  been  several 
times  re-written.  Nearly  every  instrument  mentioned  in  this 
book  is  illustrated  by  a  pretty  accurate  drawing,  which,  it  is 
hoped,  will  render  the  descriptions  intelligible  to  those  who  have 
not  the  instruments  in  their  possession. 

I  have  to  acknowledge  my  obligations  to  several  scientific 
friends  for  assistance  in  the  preparation  of  this  work.  To  my 
friends  at  Washington  and  Cambridge  I  am  indebted  for  several 
important  suggestions  ;  but  I  am  more  particularly  indebted  to 
Rev.  C.  S.  Lyman,  of  New  Haven,  who  read  nearly  the  entire 
work  in  manuscript,  and  whose  criticisms  have  proved  of  great 
service  to  me.  I  am  also  indebted  to  him  for  the  description  of 
the  prismatic  sextant  on  page  101,  and  for  the  second  method 
of  projecting  solar  eclipses  on  page  242. 

Inasmuch  as  the  student  is  supposed  to  have  some  previous 
acquaintance  with  the  elements  of  astronomy,  if  any  one  should 
undertake  the  study  of  this  volume  whose  time  does  not  permit 
him  to  read  the  whole  in  course,  he  may  take  up  whatever  chap- 
ter he  pleases,  and  omit  the  remainder  with  very  little  danger 
of  embarrassment ;  or  if  he  should  omit  any  thing  which  is  es- 
sential to  be  studied,  the  references  throughout  the  work  will 
direct  him  to  those  portions  which  require  his  attention.  To 
students  who  propose  to  devote  only  a  few  weeks  to  the  study 
of  Practical  Astronomy  as  a  branch  of  general  education,  the 
following  course  is  suggested :  Read  the  first  two  chapters,  with 
but  little,  if  any,  omission ;  read  Articles  131-5  of  Chap.  III. ; 
some  of  the  problems  of  Chap.  IV. ;  Chap.  V.  entire,  and  Chap. 
VI.  to  Art.  179  ;  a  considerable  part  of  Chap.  VIII. ,  and  Articles 
219-224  of  Chap.  IX.  ;  after  which  the  student  may  proceed 
with  Lunar  and  Solar  Eclipses  and  Occultations. 


CONTENTS. 

CHAPTER  I. 

STRUCTURE   OF   AN  OBSERVATORY.      THE  TELESCOPE. 

P»g« 

Directions  for  locating  an  Observatory 13 

Standard  Instruments  of  an  Observatory 14 

Form  of  Building  required 15 

Plan  of  Washington  Observatory 16 

The  Refracting  Telescope 17 

Positive,  negative,  and  diagonal  Eye-pieces 18 

Magnifying  Power  of  a  Telescope 19 

Mode  of  determining  the  same  experimentally 20 

To  test  the  Quality  of  a  Telescope 22 

List  of  test  Objects  for  defining  Power r^  23 

List  of  test  Objects  for  illuminating  Power 24 

List  of  remarkable  Nebulae 25 

Equatorial  Telescope  described 26 

Adjustments  of  the  Equatorial 28 

Spider-line  Micrometer  described 32 

To  find  the  Value  of  one  Revolution  of  the  Screw 34 

Position  Micrometer  described 35 

Method  of  illuminating  the  Lines 36 

Comet-seeker  described 37 

CHAPTER  II. 

THE  TRANSIT  INSTRUMENT. 

Portable  Transit  Instrument  described 39 

Transit  Instrument  for  a  large  Observatory 41 

Reversing  Stand  for  Transit  Instrument 43 

Adjustments  of  the  Transit  Instrument 44 

Properties  of  a  good  Level * 45 

Adjustment  for  Collimation 47 

Transit  Instrument  brought  into  the  Meridian 48 

An  Astronomical  Clock 51 

Method  of  observing  and  registering  Transits 52 

Equatorial  Interval  of  the  Wires 53 

To  reduce  an  Observation  when  all  the  Wires  are  not  observed 56 

Reduction  for  a  Star  near  the  Pole 57 

Reduction  for  the  Sun  or  a  Planet 59 

Reduction  of  an  imperfect  Transit  of  the  Moon's  Limb 60 

To  determine  the  Inclination  of  the  Axis  of  the  Transit 61 

To  compute  the  Correction  for  Inclination  of  the  Axis 62 

To  determine  the  Error  of  Collimation 64 


viii  CONTENTS. 

Pago 

Collimating  Eye-piece  described 66 

To  determine  the  Deviation  of  the  Transit  from  the  Meridian 68 

To  compute  the  Correction  for  Error  of  Azimuth 72 

Mode  of  testing  the  Pivots  of  the  Transit  Instrument 74 

Observations  recorded  by  means  of  Electro-magnetism 75 

The  Electric  Clock  and  Register 75 

Mode  of  reading  the  Record 78 

Personal  Equation  of  Observers 80 

CHAPTER  III. 

GRADUATED   CIRCLES. 

Mural  Circle  described 83 

Reading  Microscope  described 85 

Mode  of  making  an  Observation 86 

To  determine  the  Horizontal  Point  on  the  Limb  of  the  Circle 87 

Transit  Circle  described 89 

Differences  of  Declination  recorded  by  Electro-magnetism 92 

Altitude  and  Azimuth  Instrument  described 93 

Adjustments  of  the  Instrument 95 

Sextant  described 96 

Adjustments  of  the  Sextant 98 

Mode  of  using  the  Sextant 100 

Prismatic  Sextant  by  Pistor  and  Martins 101 

Repeating  Circle  described 103 

CHAPTER  IV. 

THE    DIURNAL    MOTION. 

The  Diurnal  Motion  described 107 

To  find  the  Altitude,  Azimuth,  and  Parallactic  Angle  of  a  Star 108 

To  compute  the  Distance  between  two  Stars Ill 

To  find  the  Altitude,  etc.,  of  a  Star  six  hours  from  the  Meridian 112 

To  find  the  Altitude,  etc.,  of  a  Star  upon  the  Prime  Vertical 113 

To  find  the  Amplitude,  etc.,  of  a  Star  when  in  the  Horizon 114 

.  Ring  Micrometer  described 115 

To  determine  the  difference  of  Right  Ascension  of  two  Stars 116 

To  determine  the  difference  of  Declination  of  two  Stars 117 

Mode  of  reducing  Comet  Observations 117 

CHAPTER  V. 

TIME. 

Solar  and  Sidereal  Day 121 

To  convert  Solar  Tune  into  Sidereal 123 

To  convert  Sidereal  Time  into  Solar 124 

To  find  the  Time  by  Equal  Altitudes  of  a  Star 126 

By  Equal  Altitudes  of  the  Sun 126 

Tables  for  facilitating  the  Computations 128 

By  a  single  Altitude  of  the  Sun  or  a  Star 132 

Time  of  Sun-rise,  allowing  for  Refraction 134 

To  determine  Time  by  the  Transit  Instrument 135 


CONTENTS.  ix 
CHAPTER  VI. 

LATITUDE. 

Latitude  by  Double  Passages  of  a  Circumpolar  Star 137     *• 

By  simple  Meridian  Altitudes 138 

By  Circum-meridian  Altitudes 140 

How  to  correct  for  Rate  of  the  Clock 144 

To  correct  for  Sun's  Change  of  Declination 146 

Latitude  by  a  single  Altitude 148 

By  Observations  of  the  Pole  Star  at  any  time  of  the  Day 149 

Latitude  by  means  of  the  Zenith  Telescope 153 

Mode  of  Observation 155 

By  Observations  with  a  Transit  Instrument  in  the  Prime  Vertical 157 

Prime  Vertical  Instrument  described 160 

Mode  of  Observation 162 

Mode  of  reducing  the  Observations 163 

To  correct  the  Result  for  Error  of  Azimuth 165 

CHAPTER  VII. 

ECLIPTIC. 

To  find  the  Position  of  the  Equinoctial  Points 168 

Right  Ascensions  corrected  by  the  Declinations 170 

To  find  the  Obliquity  of  the  Ecliptic 172 

To  find  the  Longitude  and  Latitude  of  a  Star 174 

To  find  the  Right  Ascension  and  Declination  of  a  Star 176 

To  compute  the  Longitude,  etc.,  of  the  Sun 178 

CHAPTER  VIII. 

PARALLAX. 

To  find  the  Parallax  of  the  Moon  in  Altitude 180 

Parallax  in  terms  of  the  True  Zenith  Distance 182  - 

To  find  the  Angle  of  the  Vertical 184 

To  compute  the  Radius  of  the  Earth 185 

To  find  the  Horizontal  Parallax  for  any  place 186 

To  compute  the  Moon's  Parallax  in  Right  Ascension 187 

To  compute  the  Moon's  Parallax  in  Declination 190 

Hourly  Variation  of  the  Parallax  in  Right  Ascension 196 

Hourly  Variation  of  the  Parallax  in  Declination 197 

Augmentation  of  the  Moon's  Semi-diameter 198 

CHAPTER  EC. 

MISCELLANEOUS   PROBLEMS. 

Interpolation  by  Differences 202 

Bessel's  Formula  for  Interpolation 206 

To  find  the  time  of  Conjunction  or  Opposition  of  the  Moon 209 

Hourly  Motion  of  the  Moon  in  Right,  Ascension 211 

Correction  of  the  Moon's  Decimation  in  an  Eclipse 213 

Catalogues  of  the  Fixed  Stars 21? 


x  CONTENTS. 

Page 

Correction  of  the  Mean  Places  of  the  Stars 219 

Diurnal  Aberration  of  Light 220 

Method  of  solving  Equations  of  Condition 221 

CHAPTER  X. 

ECLIPSES   OP   THE   MOON. 

Mode  of  projecting  an  Eclipse  of  the  Moon 226 

Method  illustrated  by  an  Example 228 

Mode  of  computing  the  Phases 230 

Eclipse  of  October  24,  1855 233 

CHAPTER  XI. 

ECLIPSES    OF   THE    SUN  AND   OCCULT ATIONS. 

Method  of  projecting  Solar  Eclipses 235 

Eclipse  of  May  26,  1854 237 

Method  of  Projection  by  Right  Ascension  and  Declination 242 

To  calculate  the  Beginning  and  End  of  a  Solar  Eclipse  for  any  place 247 

Formulae  of  Computation 254 

Occultations  of  Stars  by  the  Moon 258 

Bessel's  Method  of  computing  Solar  Eclipses 263 

Computation  of  the  Co-ordinates 267 

Mode  of  solving  the  Equations 273 

Points  of  first  and  last  Contact 275 

Recapitulation  of  Formulae  employed 277 

Computation  of  the  Eclipse  of  July  28,  1851 279 

Check  on  the  Accuracy  of  the  Computations 289 

Bessel's  Method  of  computing  Occultations  of  Stars 291 

Collection  of  Formulae  employed 292 

Occultation  of  a  Tauri,  January  23,  1850 293 

Check  upon  the  Accuracy  of  the  Computations 296 

Occultation  of  y  Virginis,  January  9, 1855 297 

CHAPTER  XII. 

LONGITUDE. 

Longitude  determined  by  Transportation  of  Chronometers 300 

Formulae  of  Reduction 301 

Example  from  Struve's  Chronometric  Expedition 302 

Mode  of  comparing  the  Chronometers 304 

Longitude  determined  by  the  Electric  Telegraph 305 

Mode  of  comparing  the  Clocks 306 

Longitude  of  Philadelphia  and  Hudson,  Ohio 307 

Telegraphing  Transits  of  Stars < 309 

Velocity  of  the  Electric  Fluid  determined 311 

Longitude  determined  by  Moon-culminating  Stars. 312 

Mode  of  reducing  the  Observations 313 

Imperfection  of  this  Method .  t : 316 

Longitude  determined  from  the  time  of  true  Conjunction 317 

Longitude  determined  from  the  Moon's  Motion  in  its  apparent  Orbit 320 

Longitude  from  a  Solar  Eclipse  by  Bessel's  Method 322 


CONTENTS. 


XI 


pa 

Mode  of  determining  the  Errors  of  the  Tables 326 

Collection  of  Formulae  employed 333 

Longitude  from  the  Eclipse  of  July  28,  1851 339 

Longitude  from  an  Occultation  by  Bessel's  Method 337 

Occultation  of  a  Tauri,  January  23,  1850 333 

Recapitulation  of  the  principal  Formulae  demonstrated  in  this  Work 345 

Collection  of  Trigonometrical  Formulae 352 

TABLES. 

1.  Positions  of  the  principal  Foreign  Observatories 357 

2.  Latitudes  and  Longitudes  of  places  in  the  United  States ...  358 

3.  To  convert  Hours,  etc.,  into  decimals  of  a  Day,  and  vice  versa 359 

4.  To  convert  Intervals  of  Solar  Time  into  Intervals  of  Sidereal  Time 360 

5.  To  convert  Intervals  of  Sidereal  Time  into  Intervals  of  Solar  Time 361 

6.  To  convert  Degrees  into  Sidereal  Time 362 

7.  To  convert  Sidereal  Time  into  Degrees 363 

8.  Bessel's  Refractions 364 

9.  Coefficients  of  the  Errors  of  the  Transit  Instrument 366 

10.  Reduction  to  the  Meridian 368 

11.  Equation  of  Equal  Altitudes  of  the  Sun 372 

12.  Length  of  a  Degree  of  Longitude  and  Latitude,  etc 374 

13.  Augmentation  of  the  Moon's  Semi-diameter 378 

14.  Reduction  of  the  Moon's  Equatorial  Parallax 378 

15.  Parallax  of  the  Sun  and  Planets  at  different  Altitudes 379 

16.  Moon's  Parallax  for  Cambridge  Observatory 380 

17.  Parallactic  Angles  for  Washington  Observatory 384 

18.  Correction  to  Moon's  Declination  in  computing  an  Eclipse 385 

19.  Semi-diurnal  Arcs 386 

20.  To  convert  Millimeters  into  English  Inches 388 

21.  To  convert  English  Inches  into  Millimeters 389 

22.  To  determine  Altitudes  with  the  Barometer 390 

23.  Coefficients  for  Interpolation  by  Differences 392 

24.  Logarithms  of  Bessel's  Coefficients  for  Interpolation 394 

25.  To  compare  the  Centesimal  Thermometer  with  Fahrenheit's 397 

26.  To  compare  Reaumur's  Thermometer  with  Fahrenheit's 397 

27.  Height  of  Barometer  corresponding  to  Temperature  of  boiling  Water  . . .  398 

28.  Depression  of  Mercury  in  Glass  Tubes 399 

29.  Factors  for  Wet-bulb  Thermometer 399 

30.  Catalogue  of  1500  Stars 400 

31.  Secular  Variation  of  the  Annual  Precession  in  Right  Ascension 460 

32.  Secular  Variation  of  the  Annual  Precession  in  North  Polar  Distance 461 

33.  Elements  of  the  Planetary  System 462 

34.  Elements  of  the  Satellites 463 

35.  Elements  of  the  Asteroids 464 

36.  For  Sines  and  Tangents  of  small  Arcs 466 

37.  Numbers  often  used  in  Calculations 468 

Explanation  of  the  Tables  469 

Catalogues  of  Instruments,  ivith  Prices 491 


THE  following  Alphabet  is  given  in  order  to  facilitate,  to  the  student  who  is  un- 
acquainted with  it,  the  reading  of  those  parts  in  which  the  Greek  letters  are  used : 


Letters. 

Names. 

Letters. 

Nnmes. 

A 

a 

Alpha. 

N 

V 

Nu. 

B 

P 

Beta. 

H 

sc 

Xi. 

r 

r 

Gamma. 

0 

0 

Omicron. 

A 

6 

Delta. 

n 

GT  7T 

Pi. 

E 

e 

Epsilon. 

p 

P 

Rho. 

Z 

£ 

Zeta. 

s 

Of 

Sigma. 

H 

V 

Eta. 

T 

T 

Tau. 

e 

$0 

Theta. 

T 

V 

Upsilon. 

i 

i 

Iota. 

$ 

f 

Phi. 

K 

K 

Kappa. 

X 

X 

Chi. 

A 

% 

Lambda. 

t 

j 

Psi. 

M 

P- 

Mu. 

n 

u 

Omega. 

AN    INTRODUCTION 

TO 

PRACTICAL  ASTRONOMY. 


CHAPTER  I. 

STRUCTURE  OF  AN  OBSERVATORY.— THE  TELESCOPE. 

ARTICLE  1.  In  selecting  a  site  for  an  astronomical  observatory, 
we  should  aim  to  secure  the  following  advantages : 

1.  Stability  in  the  position  of  the  instruments. 

2.  A  good  horizon. 

3.  Freedom  from  atmospheric  obstructions. 

In  order  to  secure  the  first  advantage,  we  should  select  a  spot 
which  affords  a  solid  foundation  for  building.  The  instruments 
should  rest  upon  stone  piers  whose  foundations  are  either  rock, 
gravel,  or  hard  clay,  for  which  purpose  it  is  sometimes  neces- 
sary to  excavate  the  earth  to  the  depth  of  20  or  25  feet.  To 
prevent  the  transmission  of  tremors  from  the  surface  of  the 
ground  to  the  instruments,  the  earth  should  not  be  filled  in  about 
the  piers,  but  the  latter  should  be  left  completely  insulated.  It 
is  found  that  ordinary  tremors  are  but  little  felt  a  few  feet  below 
the  surface  of  the  earth. 

Proximity  to  a  large  city  or  to  great  thoroughfares  is  most  un- 
desirable ;  but,  if  this  should  prove  unavoidable,  it  is  especially 
important  to  attend  to  the  insulation  of  the  piers. 

(2.)  In  order  to  secure  a  good  horizon,  it  was  formerly  cus- 
tomary to  build  an  observatory  of  great  height,  but,  for  the  pur- 
pose of  securing  greater  stability  of  the  instruments,  astronomers 
now  select  an  eminence  of  moderate  elevation,  from  which  the 
ground  descends  on  all  sides,  and  place  their  instruments  as  near 


14  PRACTICAL   ASTRONOMY. 

the  ground  as  can  conveniently  be  done.  It  is  a  matter  of  the 
first  importance  that  the  horizon  be  unobstructed  in  the  direction 
of  the  meridian. 

The  atmospheric  obstructions  which  astronomers  aim  to  avoid 
as  far  as  possible  are  fogs — which  are  uncommonly  prevalent  in 
certain  places,  especially  on  low,  swampy  grounds— the  smoke 
and  heated  air  arising  from  chimneys,  factories,  etc.,  as  also  the 
dust  and  noise  of  public  streets.  Certain  localities  are  much 
more  subject  to  clouds  and  high  winds  than  other  places,  and 
these  are  specially  unfavorable  to  the  operations  of  an  observ- 
atory. 

(3.)  A  transit  instrument  and  a  good  clock  are  indispensable 
to  the  furniture  of  every  observatory.  The  former  requires  an 
opening  in  the  roof  and  down  the  walls  of  the  building,  so  as 
to  afford  a  view  of  the  meridian  from  the  north  to  the  south 
horizon.  This  opening  should  not  be  less  than  eighteen  inches 
wide,  and  should  be  covered  by  doors  which  may  be  easily 
thrown  open,  and  which,  when  closed,  shall  effectually  exclude 
the  rain  and  snow.  A  complete  observatory  must  also  be  fur- 
nished with  a  graduated  circle  for  measuring  altitudes  or  polar 
distances,  which  will  require  a  second  opening  across  the  reof, 
similar  to  the  one  already  described,  unless  a  meridian  circle  be 
used  for  both  purposes,  in  which  case  one  opening  may  suffice. 

(4.)  An  altitude  and  azimuth  circle,  or  an  equatorial  instru- 
ment, requires  a  revolving  roof,  with  an  opening  from  the  zenith 
to  the  horizon,  to  enable  the  observer  to  follow  a  heavenly  body 
in  any  part  of  its  diurnal  course.  This  roof  should  not  be  larger 
than  is  necessary  for  giving  room  to  the  observer  and  to  the  in- 
strument under  it,  lest  its  bulk  and  consequent  weight  should 
impede  its  easy  motion.  It  should  be  made  to  turn  round  on 
a  circular  bed,  placed  in  a  horizontal  position.  The  dome  may 
revolve  on  small  brass  wheels,  set  in  a  ring  of  wood  of  proper 
dimensions,  or  on  cast-iron  balls,  turned  in  a  lathe  so  as  to  be  of 
exactly  equal  diameter. 

The  figure  on  the  opposite  page  represents  a  section  of  a  rota- 
tory dome  suitable  for  a  small  observatory.  The  letters  AA 
represent  an  opening  18  or  20  inches  in  width,  extending  from 
the  top  of  the  dome  down  one  side  to  the  horizon,  and  closed 
by  three,  doors,  of  which  each  upper  one  overlaps  the  next 


STRUCTURE  OF  AN  OBSERVATORY. 


15 


lower  one,  so  as  to  exclude  the  rain  and  snow.  The  wooden 
plate,  BB,  which  appears 
a  straight  line,  is  a  circu- 
lar ring,  which  forms  the 
base  of  the  dome ;  and  CO 
is  a  similar  ring,  forming 
the  wall  plate  on  which 
the  dome  rests  and  re- 
volves. 

(5.)  A  modern  observ- 
atory generally  consists 
of  a  central  building,  of 
moderate  elevation,  sur- 
mounted by  a  revolving 
dome  covering  an  equa- 
torial telescope,  and  hav- 
ing small  wings,  running 
east  and  west,  in  which 
are  placed  the  instruments  which  are  designed  for  observations 
in  the  meridian.  The  sketch  on  page  16  represents  a  section 
of  the  "Washington  Observatory.  A  is  a  pier  of  solid  masonry, 
whose  foundations  are  nine  feet  below  the  surface  of  the  ground. 
It  runs  through  the  centre  of  the  main  building,  and  on  the 
top  rests  the  equatorial,  E,  surmounted  by  &  revolving  dome. 
Both  on  the  east  and  west  sides  of  the  central  building  is  a 
wing,  each  of  which  has  two  openings  20  inches  wide,  extend- 
ing through  the  roof  and  along  the  sides  of  the  building,  so  as  to 
allow  an  unobstructed  view  of  the  meridian.  C  represents  the 
meridian  circle,  and  T  the  transit  instrument.  The  mural  circle 
was  formerly  attached  to  the  pier,  M,  in  the  west  wing,  but  it 
has  since  been  removed  to  the  pier,  P,  in  the  east  wing. 

(6.)  It  is  desirable  to  have  access  to  some  distant  field,  both 
north  and  south,  where  it  may  be  permitted  to  erect  a  pillar  on 
which  to  fix  a  meridian  mark.  This  mark  should  be  at  such  a 
distance  that  it  may  be  distinctly  seen  with  the  solar  focus  of 
the  transit  instrument,  which,  for  a  small  instrument,  may  be 
a  distance  of  half  a  mile,  but  for  a  large  instrument,  may  be  a 
mile  or  several  miles.  The  Royal  Observatory  at  Edinburgh 
has  two  meridian  marks,  the  northern  one  distant  about  8000 


16 


PRACTICAL   ASTRONOMY. 


THE    TELESCOPE. 


17 


feet,  and  the  southern  about  18,000.  These  distant  marks, 
however,  are  not  indispensable,  and  at  Greenwich  their  use  has 
been  abandoned. 

(7.)  By  applying  to  the  object  end  of  the  telescope  a  cap,  with 
a  lens  of  long  focus,  we  may  employ  a  near  meridian  mark, 
which,  in  some  ,^s?*^ 

respects,  is  more  ^^&iS^     Bfcl^^ 

convenient  than 
a  distant  one. 
The  annexed  fig- 
ure represents  a 
meridian  mark 
used  by  Captain 
Smyth,  of  Bed- 
ford,  England. 
A  brass  plate,  five  inches  long  and  three  inches  wide,  is  secured 
by  screws  to  a  stone  which  has  a  firm  foundation  sunk  into 
the  ground.  On  this  plate  there  slides  another  of  smaller  size, 
adjustable  by  two  screws  pressing  against  its  ends.  On  the 
sliding  plate  is  soldered  a  square  piece  of  silver,  bearing  a  well- 
defined  black  cross  as  a  mark  for  the  meridian.  A  four-inch 
lens,  ground  to  a  focal  length  of  49  J  feet,  which  is  exactly  its 
distance  from  the  cross,  is  attached  to  an  iron  plate,  which  is 
let  into  the  south  wall  of  the  observatory,  in  a  line  with  the 
transit  instrument.  The  rays  of  light  from  the  meridian  mark 
consequently  become  parallel  after  passing  through  the  lens, 
and  the  mark  can  be  viewed  through  a  telescope  adjusted  to  its 
solar  focus. 

THE    TELESCOPE. 

(8.)  The  object-glass  of  a  refracting  telescope  must  be  ach- 
romatic, consisting  of  two  lenses  so  combined  as  to  destroy  the 
injurious  effects  of  color  and  aberration.  The  available  diam- 
eter of  the  object-glass  is  called  its  aperture,  and  is  usually  a 
little  less  than  that  of  the  tube  in  which  it  is  inserted.  It  forms 
the  image  of  an  object  toward  which  it  may  be  directed  near 
the  eye  end  of  the  telescope.  The  distance  from  this  image  to 
the  object-glass  is  called  the  focal  length  of  the  telescope,  and 
is  commonly  a  little  greater  than  the  length  of  the  main  tube. 

B 


18  PRACTICAL   ASTRONOMY. 

This  image  is  magnified  by  a  microscope  called  an  eye-piece^ 
consisting  of  two  or  more  lenses,  and  several  of  them  are  fur- 
nished with  every  telescope,  in  order  to  afford  a  variety  of  mag- 
nifying powers.  The  eye-piece  is  set  in  a  sliding  tube,  and  is 
moved  by  a  milled  head,  connected  with  a  rack  and  pinion,  to 
enable  the  observer  to  adjust  the  eye-piece  exactly  to  the  image. 
(9.)  Two  varieties  of  eye-pieces  are  in  common  use,  one  called 
the  negative,  the  other  the  positive  eye-piece.  The  negative 
eye-piece  is  formed  of  two  plano-convex  lenses, 
A,  B,  fixed  with  their  curved  faces  toward  the 
object-glass,  at  a  distance  from  each  other 
something  less  than  half  the  sum  of  their  focal 
lengths.  It  is  called  a  negative  eye-piece,  be- 
cause  the  image  viewed  by  the  eye  is  formed 
behind  the  inner  lens,  and  this  is  the  form  generally  used  when 
distinct  vision  is  the  sole  object. 

(10.)  The  positive  eye-piece  is  formed  of  two  plano-convex 
lenses,  C,  D,  having  their  curved  faces  turned 
toward  each  other,  and  placed  at  a  distance 
from  each  other  less  than  the  focal  distance  of 
the  lens  next  the  eye,  so  that  the  image  of  the 
object  viewed  is  beyond  both  the  lenses ;  and 
this  is  the  form  adopted  for  the  transit  instrument  where  spider 
lines  are  placed  in  the  focus  of  the  object-glass,  and  also  for 
telescopes  with  micrometers,  for  the  piece  containing  the  two 
lenses  can  be  taken  out  without  disturbing  the  lines,  and  is  ad- 
justable for  distinct  vision.  As  the  image  formed  at  the  focus 
of  the  object-glass  lies  parallel  to  the  flat  face  of  the  contiguous 
lens,  every  part  of  the  field  of  view  is  distinct  at  the  same  ad- 
justment, or,  as  opticians  say,  there  is  a  flat  field. 

(11.)  In  looking  through  a  telescope  at  objects  in  high  alti- 
tudes, the  head  of  the  observer  is  brought  into  a  very  incon- 
venient position  ;  to  obviate  which  inconvenience,  the  diagonal 
eye-piece  was  invented,  and  is  commonly  applied 
to  the  transit  instrument.  A  flat  piece  of  pol- 
ished speculum  metal,  E,  is  usually  applied  be- 
tween the  two  lenses  of  the  eye-piece,  at  an  an- 
gle of  45°,  which  changes  the  direction  of  the 
rays  of  light,  and  forms  an  image  which  becomes  erect  with 


• 

THE    TELESCOPE.  19 

respect  to  altitude,  but  is  still  reversed  with  respect  to  azi- 
muth. 

(12.)  Instead  of  a  piece  of  reflecting  metal,  that  requires  a 
surface  perfectly  flat,  which  is  not  easily  obtained,  a  rectangu- 
lar prism  of  glass  is  sometimes  substi- 
tuted. A  section  of  the  prism  ABC,  per- 
pendicular to  its  edge,  must  be  an  isos- 
celes right-angled  triangle.  If,  there- 
fore, a  ray  of  light,  DE,  from  the  ob-  j> 
ject-glass  fall  upon  the  surface,  AB,  of 
the  prism  perpendicularly,  it  will  pro- 
ceed without  change  of  direction  to  E, 
will  there  suffer  total  reflection,  and  will  pass  through  the  side 
AC  without  deviation.  The  prism  has  this  advantage  over  the 
plane  speculum,  that  much  less  light  is  lost  in  the  reflection. 

(13.)  Reflecting  telescopes  are  of  various  kinds,  but  the  two 
chiefly  employed  at  present  are  the  Newtonian  and  Herschelian. 
In  the  Newtonian  form,  the  rays  reflected  from  the  large  mirror 
at  the  lower  end  of  the  tube  are  again  reflected  at  right  angles 
by  an  inclined  plane  mirror,  and  viewed  by  an  eye-piece  on  the 
side  of  the  tube.  The  observer,  accordingly,  in  using  this  in- 
strument, looks  in  a  direction  at  right  angles  with  the  tube  of 
the  telescope. 

In  the  Herschelian  construction,  the  large  mirror  is  slightly 
inclined,  so  as  to  form  the  image  close  to  one  side  of  the  tube, 
where  the  eye-piece  is  placed,  and  the  observer  looks  down  the 
tube  with  his  back  turned  toward  the  object  under  examination. 
Some  portion  of  the  light  from  the  object  is  necessarily  inter- 
cepted by  the  head  of  the  observer ;  but  in  a  large  instrument 
this  loss  is  not  very  serious. 

(14.)  If  the  solar  focal  distance  of  the  object-glass  of  the  tele- 
scope be  divided  by  the  focal  distance  of  its  eye-piece,  consid- 
ered as  a  single  lens,  the  quotient  will  express  the  magnifying 
power  of  the  telescope.  An  ordinary  celestial  eye-piece  consists 
of  two  lenses ;  so  that,  before  we  can  determine  the  magnify- 
ing power  of  the  telescope,  we  must  know  what  single  lens  is 
equivalent  to  the  two  lenses  of  the  eye-piece.  The  focal  length 

-pf 

of  the  equivalent  lens  is  given  by  the  formula  E=- — ——* 

Jb  -f/  —  a 


20  PRACTICAL   ASTRONOMY. 

where  F  denotes  the  solar  focal  length  of  the  inner,  /  that  of 
the  outer  lens,  d  the  distance  between  them,  and  E  the  focal 
length  of  the  equivalent  lens.  Then,  if  we  put  S  for  the  solar 
focal  distance  of  the  object-glass,  the  magnifying  power  will 

be8 

8  ¥ 

(15.)  As  it  is  difficult  to  measure  exactly  the  focal  length  of 
the  lenses,  other  methods  of  determining  the  magnifying  power 
of  a  telescope  are  generally  preferred. 

Let  the  focus  of  the  telescope  be  accurately  adjusted  to  dis- 
tant objects.  Then,  if  we  direct  the  telescope  toward  the  light 
of  the  sky,  a  small  bright  circle  will  be  formed  near  the  eye- 
piece, which  is  nothing  else  than  the  image  of  the  aperture  of 
the  telescope.  If,  then,  we  measure  the  diameter  of  this  circle 
by  means  of  a  scale  divided  into  very  small  equal  parts,  and  like- 
wise the  aperture  of  the  telescope,  the  diameter  of  the  aperture 
thus  determined,  divided  by  the  diameter  of  the  bright  image, 
will  express  the  magnifying  power  of  the  telescope.  For  exam- 
ple, let  the  clear  diameter  of  the  object-glass  be  10  inches,  and 
tjie  diameter  of  the  small  bright  circle  be  one  tenth  of  an  inch, 
then  will  100  represent  the  magnifying  power  of  the  telescope. 
Various  contrivances  have  been  employed  for  measuring  the  di- 
ameter of  this  small  circle  of  light,  but  the  best  method  is  by 
means  of  Ramsderfs  Dynameter. 

(16.)  The  following  is  Gauss's  method  of  determining  the 
magnifying  power  of  a  telescope :  If  we  invert  the  telescope, 
and  direct  the  eye-piece  toward  some  distant  object,  then,  on 
looking  through  the  object-glass,  the  image  of  this  object  will 
appear  as  many  times  reduced  in  size  as  it  would  be  magnified 
by  the  telescope  if  we  observed  through  the  eye-piece.  We 
therefore  direct  the  telescope  so  that  two  objects  can  be  dis- 
tinctly seen  through  the  object-glass  in  the  middle  of  the  field 
of  view,  or  at  equal  distances  on  the  two  sides  of  the  optical 
axis.  "We  then  point  a  theodolite  toward  this  telescope,  so  that 
its  optical  axis  shall  coincide  nearly  with  the  optical  axis  of  the 
telescope,  and  measure  the  angle  «,  included  between  the  ima- 
ges of  the  above-mentioned  objects  as  they  appear  in  the  invert- 
ed position  of  the  telescope.  "We  then  remove  the  telescope,  and 
measure  with  the  theodolite  the  angle  A,  which  is  comprehended 


THE  TELESCOPE.  21 
between  the  objects  themselves ;  the  required  magnifying  pow- 
er = s~ —  ;  and  if  the  angles  A  and  a  are  small,  the  mag- 

tang.  £  # 

nifying  power  =—  nearly. 

(17.)  The  magnifying  power  of  a  telescope  may  also  be  ob- 
tained in  the  following  manner :  If  a  disk  of  white  paper,  one 
inch  in  diameter,  be  placed  on  a  black  ground  at  30  or  40  yards 
distance  from  the  telescope,  and  a  staff,  painted  white  and  di- 
vided into  inches  and  parts  by  strong  black  lines,  be  placed  vert- 
ically near  the  disk,  the  eye  that  is  directed  through  the  tele- 
scope, when  adjusted  for  vision,  will  see  the  magnified  disk,  and 
the  other  eye,  looking  along  the  outside  of  the  telescope,  will  ob- 
serve the  number  of  inches  and  parts  that  the  disk  projected  on 
it  will  just  cover ;  and  the  number  of  inches  thus  covered  will 
indicate  the  magnifying  power  of  the  telescope  at  the  distance 
for  which  it  is  adjusted  to  distinct  vision. 

For  example,  a  disk  of  paper,  one  inch  in  diameter,  was 
placed  at  a  distance  of  101 J  feet,  contiguous  to  a  graduated 
vertical  staff,  and,  when  the  adjustment  for  vision  was  made 
with  a  42-inch  telescope,  the  left  eye  of  an  observer  viewed  the 
disk  projected  on  the  staff,  while  the  right  eye  observed  that  the 
enlarged  image  of  the  disk  covered  just  58 £  inches  on  the  staff; 
which  number  was  the  measure  of  the  magnifying  power  P',  at 
the  distance  answering  to  the  focal  distance  F',  which  in  this 
case  exceeds  the  solar  focal  length  F  by  an  inch  and  a  half. 
The  solar  power  P  may  be  obtained  from  the  terrestrial  or  meas- 
ured power  P'  by  the  following  proportion : 
F' :  F  :  :  P' :  P. 

In  the  present  case  we  have 

43.5  :  42  : :  58.5  :  56.5  nearly. 

Hence  the  magnifying  power  due  to  the  solar  focal  length  of 
the  telescope  is  56.5. 

(18.)  Every  telescope  of  considerable  magnifying  power 
should  be  furnished  with  a  finder ;  that  is,  a  small  telescope  of 
a  low  power  and  a  large  field  of  view,  attached  to  the  side  of 
the  larger,  with  their  axes  parallel  to  each  other.  In  the  com- 
mon focus  of  the  object-glass  and  eye-glass  is  a  pair  of  coarse 
wires,  intersecting  each  other  in  the  middle  of  the  field.  A  tel- 


22  PRACTICAL   ASTRONOMY. 

escope  with  a  high  magnifying  power  has  a  very  small  field  of 
view,  and  therefore  an  observer  may  have  great  difficulty  in 
finding  a  small  object  for  which  he  is  searching.  This  incon- 
venience is  obviated  by  the  finder.  The  telescope  is  pointed  ap- 
proximately toward  a  star  by  glancing  the  eye  along  the  tube, 
when  the  star  will  be  seen  in  the  finder,  because  its  field  of  view 
is  very  large.  The  object  is  then  brought  into  the  middle  of 
the  field  of  the  finder,  which  is  indicated  by  the  intersection  of 
the  wires,  when  it  will  be  somewhere  in  the  field  of  the  larger 
telescope. 

(19.)  In  order  to  judge  of  the  excellence  of  a  telescope,  we 
should  examine  the  quality  of  the  glass,  and  also  the  accuracy 
with  which  the  chromatic  and  spherical  aberrations  are  cor- 
rected. We  may  judge  of  the  achromatism  by  directing  the 
telescope  to  the  moon  or  to  Jupiter,  and  alternately  pushing  in 
and  drawing  out  the  eye-piece  from  the  place  of  distinct  vision. 
In  the  former  case,  a  ring  of  purple  will  be  formed  round  the 
edges ;  and  in  the  latter,  a  ring  of  light  green,  which  is  the  cen- 
tral color  of  the  prismatic  spectrum ;  for  these  appearances  show 
that  the  extreme  colors,  red  and  violet,  are  corrected. 

(20.)  "We  may  test  the  figure  of  the  object-glass  by  covering 
its  centre  by  a  circular  piece  of  paper,  about  one  half  of  its  di- 
ameter, and  adjusting  it  for  distinct  vision  of  a  given  object,  and 
then  trying  if  the  focal  length  remains  unaltered  when  the  pa- 
per is  taken  away,  and  a  cap  with  an  aperture  of  the  same  size 
is  applied,  so  that  the  extreme  rays  may  in  their  turn  be  cut  off. 
If  the  vision  is  distinct  in  both  cases,  without  any  new  adjust- 
ment for  focal  distance,  the  spherical  aberration  is  corrected. 

(21.)  If  one  part  of  the  object-glass  have  a  different  refractive 
power  from  another  part,  a  star  of  the  first  magnitude  will  point 
out  the  defect  by  the  exhibition  of  an  irradiation,  or  what  op- 
ticians call  a  wing,  at  one  side,  which  no  perfection  of  figure 
or  of  adjustment  will  banish  ;  and  the  greater  the  aperture,  the 
more  liable  is  the  evil  to  happen.  Hence  caps  with  different 
apertures  are  usually  supplied  with  large  telescopes,  that  the 
extreme  parts  of  the  glass  may  be  cut  off  in  observations  requir- 
ing a  well-defined  image.  In  case  one  half  of  the  glass  be  faulty 
and  the  other  good,  a  semicircular  aperture,  by  being  turned 
gradually  round,  will  detect  the  semicircle  which  contains  the 


THE    TELESCOPE. 


23 


defective  portion  of  the  glass  ;  and  if  such  portion  should  be  cov- 
ered, the  only  inconvenience  that  would  ensue  would  be  the  loss 
of  the  light  which  is  thus  excluded. 

(22).  The  most  precise  mode  of  estimating  the  capacity  of  a 
telescope  is  by  observations  of  a  series  of  test  objects  in  the 
heavens.  These  objects  should  be  selected  with  reference  both 
to  illuminating  power  and  denning  power,  which  qualities  are 
quite  distinct  from  each  other.  The  usual  tests  of  illuminating 
power  are  stars  of  such  a  degree  of  faintness  as  barely  to  corne 
within  the  range  of  the  telescope ;  and  the  tests  of  defining 
power  are  double  stars,  as  close  to  each  other  as  can  be  dis- 
tinctly seen  separated.  The  following  list  of  close  double  stars 
will  afford  a  considerable  range  of  tests  for  defining  power : 


Star.                         |R.A.,1850. 

Dec.,  1850. 

Magnitudes. 

Distance. 

7)  Herculis 

h.    m.    s. 
16  37  45 
15  36  27 
13     2  41 
1  54  43 
9  20  25 
20  43  28 
20  24     2 
15  18  51 
20  51  35 
16  55  41 
3  25  34 
15  17     0 
2  50  39 
17  54  55 
5  22  46 
0  46  57 
22  22  23 
8     3  36 
16  35  38 
14  33  59 
7  32     9 
16  23  21 
18     2     7 
19  41  38 
13  26  38 
16     9     4 
11   16     6 
18  39  25 
18  39  22 
2  35  32 
6  21  33 
11   10  10 
10  11  42 
12  34    4 
14  38  26 
19  48  39 
17     2  14 
1  54  18 

39    12.  7  N. 
26  46.5  N. 
18   19.5  N. 
41   36.5  N. 
9  42.4  N. 
6  11.0  S. 
10  45.5  N. 
37  52.6  N. 
3  43.3  N. 
65   16.1  N. 
23  57.5  N. 
30  50.0  N. 
20  44.2  N. 
8  10.5  S. 
5  49.8  N. 
22  48.9  N. 
3  40.3  N. 
18     5.8  N. 
31  52.7  N. 
14  22.5  N. 
5  34.3  N. 
2   19.0  N. 
3  58.4  N. 
11  26.8  N. 
0  27.3  N. 
34   14.5  N. 
11  21.3  N. 
39  27.5  N. 
39  30.9  N. 
2  36.1  N. 
6  56.4  S. 
32  22.4  N. 
20  35.9  N. 
0  37.6  S. 
27  42.6  N. 
69  53.1  N. 
54  40.1  N. 
2     2.3  N. 

A3     B8 
A5     B7 

A  4^  B5 
B  5fr  C  6 
A6^  B  7 
A  6     B7^ 
B8     C9 
A  8     B  8i 
A5fr  B7 
A  7     B  7i 
A  6     B6| 
A  6     B  6i 
A  6     B  7 
A  6     B  6i 
A  5     B  7 
A  6     B  7 
A  6     B7| 
A  6     B  6fc 
A3     B  7 
A4     B  4| 
A7     B8 
A4     B  6 
A  6     B  7^ 
A6     B  7 
A  8     B  9 
A6     B7 
A4    B  8 
C  5J-  D  5^ 
A  5^  B  6* 
A3     B  7 
B  7     C8 
A  4£  B  5i 
A2£  B4 
A4     B4 
A3     B  7 
A5£  B9 
A  5     B  5 
A5     B6 

0.3 
0.4 
0.4 
0.4 
0.4 
0.5 
0.6 
0.6 
0.7 
0.7 
0.7 
0.8 
0.8 
0.9 
1.0 
1.1 
1.1 
1.2 
L2 
1.2 
1.3 
1.4 
1.5 
1.7 
1.7 
2.2 
2.5 
2.6) 
3.2  ( 
2.6 
2.7 
2.7 
2.9 
3.0 
3.0 
3.1 
3.2 
3.7 

Binary. 
Quadruple. 

Binary. 
Binary. 

Triple  star. 
Binary. 

Binary. 

Binary. 
Binary. 

Multiple. 

Triple  star. 
Binary. 

Binary. 

'  Coronse  Borealis  
42  Comae  Berenicis 

•y2  Andromeda?  
d)  Leonis 

Ir  Aquarii  
178  P.  Delphini  
LI*  Bootis 

£  Equulei  
20  Draconis  

7  Tauri 

i]  Coronse  Borealis  
e  Arietis  
T  Ophiuclii  
32  Orionis  

36  Andromedse  
37  Pegasi 

^  Cancri 

£  Herculis  

£  Bootis 

170  P.  Canis  Minoris  
/I  Ophiuclii  

73  Ophiuchi  
TT  Aquilse  
127  P  Virginis 

a  Coronze  Borealis  
i  Leonis 

e2  Lyrse  
£l  Lyrse  

y  Ceti   . 

1  1  Monocerotis  
£  Ursa}  Majoris 

y  Leonis  
y  Virginis 

e  Bootis  
£  Draconis 

ft  Draconis  

a  Piscium  

24 


PRACTICAL   ASTRONOMY. 


(23.)  As  tests  of  illuminating  power  may  be  mentioned  the 
satellite  of  Neptune,  which  is  equal  to  a  star  of  the  fourteenth 
magnitude,  and  is  not  known  to  have  been  seen  by  more  than 
three  or  four  telescopes  in  any  part  of  the  world ;  the  satellites 
of  Uranus,  which  are  about  equally  difficult ;  and  the  three 
smaller  satellites  of  Saturn.  The  new  satellite  of  Saturn,  dis- 
covered by  Mr.  Bond,  is  estimated  to  be  equal  to  a  star  of  the 
seventeenth  magnitude. 

The  following  table  of  faint  and  unequal  stars  will  also  afford 
a  good  measure  of  illuminating  power : 


Star. 

R.  A.,  1850. 

Dec.,  1850. 

Magnitudes. 

Distance. 

d  Cygni  

h.    m.     s. 
19  40    17 

0              / 

44  46.0  N. 

A3     B    9 

1.7 

y  Crateris  
o  Cassiopese  .  .  . 
1  1  Cancri  
Antares  

11    17  23 
23  51   26 
7  59  39 
16  20  13 

16  51.6  S. 
54  55.2  N. 
27  54.6  N. 
26     5.6  S. 

A  4     B  14 
A6     B    8 
A  7     B  12 
A  1     B  10 

3.0 
3.1 
3.2 
3.5 

17  Lyrae  
84  Ceti  
a2  Capricorn!  .  .  . 
0  Virginis  
15  Pleiadum  .  .  . 
v  Ceti  

19     1  45 
2  33  33 
20     9  43 
14  20  29 
3  36  58 
2  28     0 

32  16.1  N. 
1  20.2  S. 
13     0.3  S. 
1  33.1  S. 
22  40.5  N. 
4  56.1  N. 

A  6     B  11 
A  6     B  14 
A3     B  16 
A  5     B  13 
A  8     B  14 
A  4J-  B  15 

3.6 

5.0 
5.0 
5.0 
5.0 
6.0 

K  Geminorum  .  . 
34Piscium  ..«.'. 
(J  Geminorum  .  . 
^  Piscium  
0  Orionis  
77  Cassiopese  .... 
A  Geminorum  .  . 
6  Persei  
41  Arietis  
£  Pegasi  
T  Orionis  
Polaris  .  . 

7  35  23 

0     2  20 
7  11   10 
1     5  37 
5     7  19 
0  40     3 
7     9  28 
2  33  58 
2  41     9 
22  39   12 
5  10  19 
151 

24  45.2  N. 
10  18.6  N. 
22  15.2  N. 
23  47.3  N. 
8  22.8  S. 
57     1.1  N. 
16  48.4  N. 
48  35.4  N. 
26  38.4  N. 
11  24.4  N. 
7     0.6  S. 
88  30  6  N 

A  4     B  10 
A  6     B  13| 
A4    B    9 
A  6     B  13 
A  1     B    9 
A  4    B    7£ 
A4    B  12 
A4    B  13 
A3     B  13 
A  5     B  15 
A  4     B  15     C  12 
A  2£  B    9£ 

6.1 
7.0 
7.1 
9.0 
9.5 
9.5 
10.3 
15.0 
15.0 
15.0 
15.0  and  20.0 
18  6 

ft  Aquarii  
7  Camelopardi  .  . 
6  Equulei  
42  Piscium  
a  Lyrae  

21  23  39 
4  45  16 
21     7  11 
0  14  40 
18  31   51 

6  13.7  S. 
53  30.3  N. 
9  24.1  N. 
12  38.9  N. 
38  38  8  N 

A3.    B  15 
A5     B  13 
A  4^  B  11 
A7     B  13 
A  1     B  11 

25.0 
27.0 
28.2 
35.0 
434 

a  Arietis  .  . 

1  58  44 

22  45.0  N. 

A3    B  11 

300 

(24.)  'Some  of  the  nebulae  afford  excellent  tests  of  the  per- 
formance of  a  telescope.  Certain  nebulae,  which,  to  an  ordinary 
telescope,  appear  merely  as  a  dim  patch  of  light,  by  more  pow- 
erful instruments  are  resolved  wholly  into  stars,  and  others  are 
partially  resolved. 

The  following  are  some  of  the  most  interesting  objects  of 
this  class : 


THE    TELESCOPE. 


25 


|R.A.,1850. 

Dec.,  1850. 

Nebula  of  Andromeda  
33  H  Persei                   

ft.   m.   s. 
0  34  37 
2     8  39 
2  13  12 
3  58  55 
5  25  27 
5  27  54 
6  46  31 
8  42  58 
10  17  29 
11     5  59 
12  36     3 
12  49  21 
13     5  32 
13  23  31 
13  35  12 
14  21  44 
15  10  56 
16  36   19 
16  49   16 
18   11   57 
18   15  17 
18  47  59 
19  10  42 
19  53     4 
20  26  50 
21  22  42 
21  25  40 
21  31  50 

40  26.9  N. 
56  27.1  N. 
41  38.9  N. 
49     6.2  N. 
21  54.7  N. 
5  29.5  S. 
18     9.8  N. 
12  21.4  N. 
17  53.6  S. 
55  49.7  N. 
12  22.8  N. 
22  30.0  N. 
18  58.1  N. 
47  58.6  N. 
29     7.3  N. 
5  17.8  S. 
2  39.0  N. 
36  44.6  N. 
3  52.8  S. 
16  15.8  S. 
24  56.7  S. 
32  50.8  N. 
29  55.2  N. 
22  18.7  N. 
6  55.2  N. 
11   30.0  N. 
1  29.5  S. 
23  49.8  S. 

Large  and  irresolvable. 
Glorious  mass  of  stars. 
Irresolvable. 
Compressed  oval  group. 
Large  oval  nebula. 
Great  irresolvable  nebula. 
A  compressed  cluster. 
Loose  cluster. 
Fine  planetary  nebula. 
Planetary  nebula. 
A  double  nebula. 
Not  resolved. 
A  globular  cluster. 
Irresolvable  pair  of  nebulae. 
A  globular  cluster. 
Resolved  in  largest  telescopes. 
A  mass  of  stars. 
A  splendid  cluster. 
A  rich  round  mass. 
Horse-shoe  nebula. 
Compact  globular  cluster. 
Annular  and  irresolvable. 
Globular  cluster. 
Irresolvable. 
A  mass  of  minute  stars. 
Globular  cluster. 
A  ball  of  stars. 
A  bright  cluster. 

19  H  Andromedse  

60  H  Persei         

Nebula  in  Orion  

2  H.  Geminorum  
67  M  Cancri 

27  H  Hydras  

97  M.  Ursse  Majoris  

60  M.  Virginis  
64  M.  Comas  Berenicis  .... 
53  M.  Comae  Berenicis  ,  .  .  . 
Spiral  nebula      

3  M.  Canum  Venaticorum  . 
70  H.  Virginis      

5  M  Libra 

13  M.  Herculis  

10  M.  Ophiuchi  

17  M.  Clypei  Sobieskii  
28  M.  Sagittarii  
Annular  nebula  in  Lyra  .  .  . 
56  M.  Lyrae  
Dumb-bell  nebula  

103  H.  Delphini   
15  M.  Pegasi  
2  M.  Aquarii 

30  M.  Capricorni  

Many  of  the  preceding  nebulae  are  quite  conspicuous,  even 
with  a  small  telescope  ;  but  the  visible  boundaries  of  these  ob- 
jects, and  the  number  of  stars  which  they  exhibit,  depend  upon 
the  power  of  the  instrument. 

(25.)  The  following  statement,  by  Captain  Smyth,  will  afford 
to  beginners  a  tolerable  idea  of  what  kind  of  performance  they 
ought  to  expect  from  their  telescopes.  Captain  Smyth's  tele- 
scope was  an  achromatic  refractor  of  8J  feet  focal  length,  with 
an  object-glass  of  5T9^  inches  clear  aperture.  The  cap  which 
covered  the  object-glass  was  pierced  with  two  circular  holes,  of 
two  inches  and  four  inches  diameter.  "With  the  two-inch  aper- 
ture, and  magnifying  powers  of  from  60  to  100,  he  saw  Polaris 
and  its  companion  distinctly,  and  clearly  perceived  double 


a  Piscium, 
j  Leonis, 


\i  Draconis, 
7j  Cassiopese. 


With  the  four-inch  aperture,  and  powers  varying  from  80  to 
120,  and  upward,  he  readily  saw 


26  PRACTICAL    ASTRONOMY. 


j3  Orionis, 
a  Lyrse, 
6  Greminorum, 
£  Ursae  Majoris, 


0  Cassiopese, 
y  Ceti, 

e  Draconis, 

1  Leonis. 


But  it  required  the  full  aperture,  and  powers  of  from  240  to 
300,  with  favorable  circumstances,  to  scrutinize  satisfactorily 
the  following  test  objects : 


a  Arietis, 
A  Ophiuchi, 
£  Equulei, 
6  Cygni, 


20  Draconis, 
£  Herculis, 
32  Orionis, 
K  G-eminorum. 


EQUATORIAL    TELESCOPE. 

(26.)  An  equatorial  telescope  consists  of  a  telescope  so  mount- 
ed as  to  have  two  axes  of  motion  at  right  angles  to  each  other, 
and  also  a  graduated  circle  connected  with  each  axis,  at  right 
angles  to  its  length.  "When  the  instrument  is  adjusted  for  use, 
one  axis  is  parallel  to  the  axis  of  the  earth,  and  is  called  the 
polar  axis  ;  the  other  is  parallel  to  the  plane  of  the  equator,  and 
is  called  the  declination  axis.  When  a  telescope  is  mounted 
with  an  altitude  and  azimuth  movement  (as  is  the  case  with 
common  portable  instruments),  it  requires  a  motion  in  altitude 
as  well  as  in  azimuth  to  follow  a  star  in  its  diurnal  course ; 
and  these  movements  are  sufficient  to  occupy  both  hands  of  the 
observer.  But  with  a  telescope  mounted  equatorially,  only  one 
motion  is  required  to  follow  a  star  ;  that  is,  a  motion  parallel  to 
the  plane  of  the  equator ;  and  this  motion  being  perfectly  uni- 
form, can  easily  be  effected  by  clock-work ;  by  which  means  the 
observer  has  both  his  hands  at  liberty  to  use  a  micrometer,  or 
for  any  other  purpose.  Such  an  instrument,  therefore,  affords 
great  advantages  in  measuring  the  relative  position  of  two  con- 
tiguous bodies,  in  measuring  the  diameters  of  the  planets,  etc. 
The  circle  which  is  connected  with  the  polar  axis  is  graduated 
into  hours,  minutes,  and  seconds  of  time,  to  indicate  the  right 
ascension  of  the  object  under  examination ;  while  the  circle 
connected  with  the  declination  axis  is  graduated  into  degrees, 
minutes,  and  seconds  of  arc,  to  indicate  declination  or  polar 
distance. 


EQUATORIAL    TELESCOPE. 


(27.)   The  mode  of  mounting  now  generally  preferred  is  that 
employed   by  Fraun- 
hofer,    and    is    repre- 
sented in  the  annex- 
ed cut. 

a  represents  the 
polar  axis  parallel  to 
the  axis  of  the  earth ; 
b  is  the  right  ascen- 
sion circle  attached  to 
the  supporting  frame, 
while  two  verniers, 
attached  to  the  polar 
axis,  and  revolving 
with  the  telescope, 
point  out  the  right  as- 
cension of  a  star  upon 
this  circle.  The  axis 
c  is  the  declination 
axis,  at  right  angles 
with  the  polar  axis, 
and  is  mounted,  so  as 
to  revolve  in  its  sup- 
ports ;  while  the  dec- 
lination circle,  d,  in- 
dicates the  declina- 
tion of  the  object 
under  examination. 
When  the  right  ascension  circle  is  clamped,  the  declination  axis 
may  be  made  to  revolve  through  360°,  by  which  means  the 
telescope  will  be  pointed  successively  to  every  degree  of  dec- 
lination. "When  the  declination  circle  is  clamped,  the  polar 
axis  may  be  made  to  revolve  through  360°,  by  which  means 
the  telescope  will  describe  a  complete  circle  of  diurnal  motion. 
Thus,  by  means  of  these  two  motion s, 'which  are  at  right  an- 
gles to  each  other,  the  telescope  may  be  turned  toward  any  part 
of  the  heavens.  Indeed,  there  are  always  two  positions  of  the 
instrument,  with  reference  to  the  polar  axis,  in  which  the  tel- 
escope may  be  pointed  upon  any  star.  If  we  suppose  the  tele- 


28  PRACTICAL   ASTRONOMY. 

scope  to  be  in  the  position  represented  in  the  preceding  cut, 
and  revolve  it  180°  in  right  ascension,  and  also  about  180°  in 
declination,  the  telescope  will  point  toward  the  same  part  of  the 
heavens  as  at  present ;  but  the  telescope  will  be  on  the  west 
side  of  the  polar  axis  instead  of  the  east  side.  One  is  called 
the  direct  position ;  the  other,  the  reversed  position  of  the  tel- 
escope. 

Connected  with  the  polar  axis  is  clock-work,  represented  at 
e,  by  which  the  instrument  is  turned,  so  as  to  follow  the  diurnal 
motion  of  a  star,  without  the  necessity  of  any  interference  from 
the  observer.  The  driving  power  is  the  descent  of  a  weight, 
g,  which  communicates  motion  to  a  train  of  wheel- work,  and 
ultimately  to  the  polar  axis,  while  its  too  swift  descent  is  reg- 
ulated by  the  friction  of  centrifugal  balls.  This  contrivance 
serves  to  retain  any  object  upon  which  the  telescope  may  be 
pointed  in  the  centre  of  the  field  of  view  for  hours  in  succession, 
leaving  the  attention  of  the  observer  undistracted,  and  both  his 
hands  at  liberty. 

(28.)   The  equatorial  requires  the  following  adjustments  : 

1.  The  polar  axis  must  be  elevated  to  the  altitude  of  the 
pole. 

2.  The  index  of  the  declination  circle  must  point  to  zero 
when  the  line  of  collimation  is  parallel  to  the  equator. 

3.  The  polar  axis  must  be  brought  into  the  meridian. 

4.  The  line  of  collimation  of  the  telescope  must  be  perpen- 
dicular to  the  declination  axis. 

5.  The  declination  axis  must  be  perpendicular  to  the  polar 
axis. 

6.  The  index  of  the  hour  circle  must  point  to  zero  when  the 
telescope  is  in  the  meridian  of  the  place. 

(29.)  First  Adjustment. — Observe  the  polar  distance  of  any 
known  star  when  near  the  meridian,  and  then,  turning  the 
polar  axis  half  round,  observe  the  same  star  again.  Take  the 
mean  of  the  two  observations,  which  is  the  distance  of  the  star 
from  the  pole  of  the  instrument ;  correct  it  for  refraction,  and 
compare  the  result  with  the  true  north  polar  distance  given  by 
the  Nautical  Almanac.  If  the  star  is  above  the  pole,  and  the 
instrumental  exceeds  the  true  polar  distance,  the  pole  of  the 
instrument  is  below  the  pole  of  the  heavens,  and  vice  versa. 


EQUATORIAL    TELESCOPE,  29 

Correct  this  error  by  the  proper  screws  for  raising  or  depressing 
the  polar  axis. 

Example.  When  e  Ursse  Minoris  was  near  the  meridian,  its 
north  polar  distance  was  observed  to  be  7°  44'  7",  the  face  of 
the  declination  circle  being  west ;  and  7°  44'  40"  when  the  face 
of  the  circle  was  east. 

The  mean  of  these  two  observations  is  7°  44'  23".5;  the 
refraction  was  52". 8  ;  making  the  corrected  polar  distance 
7°  43'  30".7.  The  polar  distance  by  the  Nautical  Almanac 
was  7°  42'  40".7.  Hence  the  polar  axis  was  50"  too  low.  The 
refraction  is  derived  from  Table  VIII.,  page  364. 

(30.)  Second  Adjustment. — Take  half  the  difference  of  the 
above  two  observations ;  this  will  be  the  index  error  of  the 
declination  verniers,  and  they  must  be  moved  accordingly  by 
their  adjusting  screws.  Several  pairs  of  observations  should  be 
taken,  in  order  to  ascertain  these  errors  with  great  accuracy. 

Example.  According  to  the  observations  above  given,  the 
index  error  was  16".5,  to  be  added  to  observations  when  the 
circle  is  west. 

(31.)  Third  Adjustment. — Observe  the  polar  distance  of  a 
star  which  is  six  hours  from  the  meridian,  the  star  being  not 
very  near  the  pole,  nor  yet  near  the  horizon.  Correct  this  for 
refraction  in  polar  distance,  and  compare  the  result  with  the 
true  polar  distance  from  the  Nautical  Almanac.  If  the  star  is 
to  the  east  of  the  meridian,  and  the  instrumental  exceeds  the 
apparent  polar  distance,  the  north  pole  of  the  instrument  is  to 
the  west  of  the  celestial  pole. 

Example.  The  polar  distance  of  a  Ursse  Majoris,  when  six 
hours  west  from  the  meridian,  was  observed  to  be  27°  23'  49".0r 
the  face  of  the  circle  being  west.  Correcting  this  for  index  er- 
ror found  above,  16".5,  and  for  refraction  30".8,  the  result  is  27° 
24'  36".3.  The  polar  distance  by  the  Nautical  Almanac  is  27° 
24'  43".7.  Hence  the  pole  of  the  instrument  was  7".4  west. 

(32.)  The  influence  of  refraction  upon  the  right  as- 
cension and  declination  of  a  star  may  be  computed  as 
follows : 

Let  A  represent  the  true  position  of  a  star,  B  its 
apparent  position  affected  by  refraction ;  then  AB  rep- 
resents the  refraction  in  altitude.  Let  BC  represent 


30  PRACTICAL   ASTRONOMY. 

a  portion  of  an  hour  circle  passing  through  B,  and  let  AC  be  an 
arc  perpendicular  to  BC.  Then  ABC  may  be  regarded  as  a  plane 
right-angled  triangle,  in  which  AC  represents  the  effect  of  re- 
fraction in  right  ascension,  and  BC  the  effect  in  declination. 

Now     AC  =  AB  sin.  ABC  ;  and  BC  =  AB  cos.  ABC. 

The  angle  ABC,  which  we  will  represent  by  p,  is  called  the 
parallactic  angle,  and  the  mode  of  computing  it  is  shown  in 
Art.  145.  Hence  the  refraction  in  right  ascension  is  equal  to 
the  refraction  in  altitude,  multiplied  by  the  sine  of  the  paral- 
lactic angle ;  and  the  refraction  in  declination  is  equal  to  the 
refraction  in  altitude,  multiplied  by  the  cosine  of  the  parallactic 
angle.  The  refraction  in  right  ascension  is  here  expressed  in 
parts  of  a  great  circle ;  if  we  wish  to  reduce  it  to  arc  of  right 
ascension,  we  must  divide  this  result  by  the  cosine  of  the  star's 
declination,  as  shown  in  Art.  72.  Henc6  we  have 

-r,   .       .      •     T>     *       ref-  m  alt-  x  sm-  P 
Refraction  in  R.  A.  —  —  — -. 

cos.  dec. 

Refraction  in  Dec.  — ref.  in  alt.  x  cos.  p. 

(33.)  Fourth  Adjustment. — Observe  the  transit  of  an  equa- 
torial star  over  the  middle  vertical  wire,  or  mean  of  the  wires  ; 
note  the  time,  and  read  off  the  verniers  of  the  hour  circle. 
Turn  the  polar  axis  half  round,  and  observe  the  same  star  a 
second  time  exactly  as  before.  Now  the  interval  between  the 
two  observations  should  correspond  exactly  to  the  difference  be- 
tween the  two  readings  of  the  hour  circle.  If  they  do  not  cor- 
respond, it  is  evident  that  one  of  the  transits  has  been  observed 
too  early,  and  the  other  too  late,  on  account  of  the  erroneous  po- 
sition of  the  wires.  One  half  the  difference  between  the  inter- 
val as  measured  by  the  clock,  and  that  by  the  hour  circle,  will 
be  the  error  of  collimation. 

Example.  The  following  observations  were  made  upon  6 
Ophiuchi : 


Face  of  Circle. 

Sidereal  Tune. 

Hour  C'irclc. 

West     .     .     . 
East      .     .     . 

16  12  58.8 
16  19     9.7 

h.      m.         s. 

0     6  51.0 
0  13     0.5 

The  interval  between  the  two  observations  is  6m.  10.9s. ; 
the  difference  between  the  two  readings  of  the  hour  circle  is 
6m.  9.5s.  One  half  the  difference  is  0.7s.,  which  is  the  error 


EQUATORIAL  TELESCOPE.         31 

of  collimation  to  be  added  to  the  readings  of  the  hour  circle 
when  the  circle  is  east. 

(34.)  Fifth  Adjustment.  —  The  decimation  axis  should  "be 
placed  by  the  maker  perpendicular  to  the  polar  axis,  and,  hav- 
ing been  once  accurately  adjusted,  is  not  liable  to  subsequent 
derangement.  The  accuracy  of  this  adjustment  may  be  tested 
as  follows  :  Bring  the  declination  axis  into  a  horizontal  position 
by  means  of  a  spirit-level,  whose  legs  rest  upon  the  extremities 
of  the  declination  axis,  and  read  the  hour  circle.  Turn  the  po- 
lar axis  half  round ;  bring  the  declination  axis  into  a  horizontal 
position  by  means  of  the  level,  as  before,  and  again  read  the 
hour  circle.  If  the  readings  agree  in  both  positions,  or  differ  by 
12h.  (when  the  graduation  is  to  24h.),  the  declination  axis  is 
adjusted.  If  the  readings  do  not  agree,  the  declination  axis  is 
*iot  perpendicular  to  the  polar  axis.  If  the  declination  axis  is 
furnished  with  adjusting  screws,  place  the  hour  circle  half  way 
between  the  position  it  actually  has  and  that  which  it  ought  to 
occupy,  in  order  that  the  readings  may  differ  by  exactly  twelve 
hours,  and  make  the  declination  axis  horizontal  by  raising  or 
depressing  the  proper  screws. 

(35.)  This  adjustment  may  be  tested  astronomically  as  fol- 
lows :  Observe  the  transit  of  a  star  not  less  than  45°  from  the 
equator,  in  both  positions  of  the  polar  axis,  as  directed  for  the 
fourth  adjustment.  Since  an  elevation  of  the  west  end  of  the 
declination  axis  causes  the  line  of  sight  to  describe  a  circle  to 
the  east  of  the  pole,  all  the  transits  observed  in  that  position 
will  be  too  early ;  and,  vice  versa,  all  will  be  too  late  when  the 
east  end  is  high.  Again,  if  the  west  end  is  too  high  before 
reversing,  the  east  end  is  too  high  after  reversing,  so  that  an 
error  of  inclination  has  a  different  effect  upon  observations  in 
reversed  positions,  and  thus  the  interval  is  increased  or  dimin- 
ished by  twice  the  error  of  a  single  observation.  The  law  of  the 
error  is,  that  it  varies  as  the  tangent  of  the  star's  declination. 
If  we  represent  the  interval  between  the  observations,  as  meas- 
ured by  the  clock,  by  c,  and  the  interval,  as  measured  by  the 
hour  circle,  by  A,  then 

c-h 

2  tang.  dec. 
will  be  the  error  in  the  position  of  the  declination  axis. 


32  PRACTICAL   ASTRONOMY. 

(36.)  Sixth  Adjustment. — Set  the  declination  axis  horizontal 
"by  means  of  the  level,  when,  if  the  previous  adjustments  have 
been  properly  performed,  the  instrument  will  be  in  the  meridian, 
and  the  verniers  may  be  set  to  zero.  Or,  clamp  the  instrument 
approximately  in  the  meridian,  observe  the  transit  of  one  or 
more  known  stars  not  far  from  the  equator,  and  correct  the  time 
of  observation  for  the  error  of  the  clock.  Then,  since  the  right 
ascension  of  the  star = the  true  sidereal  time  of  observation,  ± 
the  true  hour  angle  from  the  meridian,  the  true  hour  angle  is 
known,  and  the  verniers  may  be  set  to  mark  it. 

If  it  is  proposed  to  determine  the  absolute  place  of  a  heaven- 
ly body  by  means  of  the  equatorial,  it  is  necessary  to  determine 
its  errors  with  great  accuracy ;  but  this  instrument  is  chiefly 
employed  for  determining  differences  of  right  ascension -and  dec- 
lination of  objects  very  near  each  other,  in  which  case  entire  ac-» 
curacy  in  all  the  adjustments  becomes  oJB^ss  importance. 

THE    MICROMETER. 

(37.)  The  object  of  the  micrometer  is  to  measure  small  ce- 
lestial arcs  in  the  field  of  view  of  a  telescope.  It  appears  un- 
der a  great  variety  of  forms ;  but  the  one  now  most  commonly 
employed  is  called  the  spider-line  micrometer,  or  filar  microm- 
eter. It  consists  essentially  of  two  parallel  spider  lines  inserted 
in  the  common  focus  of  the  object-glass  and  eye-glass,  in  such  a 
manner  that  they  may  be  made  to  coincide  or  be  separated  by 
the  slow  motion  of  a  screw.  The  number  of  revolutions  and 
parts  of  a  revolution  of  the  screw  are  indicated  by  a  scale  out- 
side of  the  tube,  and  this  affords  a  measure  of  the  distance  of 
the  spider  lines,  or  of  any  two  celestial  objects  with  which  they 
may  be  made  to  coincide. 

(38.)  The  figure  on  the  opposite  page  represents  the  spider- 
line  micrometer,  as  made  by  Troughton.  It  consists  of  a  rect- 
angular box,  three  or  four  inches  long,  about  one  inch  broad, 
and  a  quarter  of  an  inch  in  thickness,  with  a  graduated  screw- 
head  at  each  extremity ;  aaaa  are  the  sides  of  the  box  seen 
edgewise  ;  bbb,  ccc  are  two  forks  of  brass,  which  slide  one 
within  the  other,  in  opposite  directions,  and  across  them  are 
stretched  the  spider  lines  d  and  e ;  ff  are  fine  screws  attached 
to  the  forks,  and,  passing  through  the  ends  of  the  box,  enter 


THE    MICROMETER. 


33 


the  milled  heads  gg,  with  each  of  which  is  connected  a  small 
graduated  circle.     Whenever  the  heads  gg  are  turned  in  the 


direction  of  the  numbers  upon  the  circles,  the  forks  be  are 
drawn  outward  ;  and  when  they  are  turned  in  the  contrary  di- 
rection, the  springs  hh  push  the  forks  inward,  and  thus  prevent 
any  loss  of  motion  in  the  screw.  The  screws  have  about  100 
threads  to  the  inch,  so  that  one  revolution  of  the  head  g  car- 
ries the  line  d  over  the  hundredth  part  of  an  inch.  The  cir- 
cumference of  the  circle  attached  to  the  head  g  is  divided  into 
a  hundred  equal  parts,  so  that  the  motion  of  the  head  g  through 
one  of  these  divisions  advances  the  line  d  through  one  ten  thou- 
sandth part  of  an  inch.  The  long  line  /,  running  at  right  angles 
to  the  small  ones,  should  be  placed  parallel  to  the  two  objects 
whose  distance  is  to  be  measured. 

On  one  side  of  the  field  of  view  is  a  notched  scale  of  teeth, 
corresponding  in  size  to  the  threads  of  the  screw.  Every  fifth 
one  is  cut  deeper  than  the  rest,  and  they  are  numbered  from 
zero,  at  the  centre,  by  tens,  in  each  direction.  The  spider  lines 
may  be  made  to  coincide  at  zero,  which  is  represented  by  the 
small  circular  hole  made  near  the  middle  of  the  scale,  and  they 
may  be  made  to  glide  by  each  other  a  short  distance,  e.  passing 
very  close  under  d. 

(39.)  In  order  to  measure  any  distance,  as,  for  example,  the 
sun's  diameter,  turn  one  of  the  heads  until  the  attached  line  is 
drawn  15  or  20  notches  to  the  left  of  zero,  and  the  other  head 
in  the  contrary  direction,  until  d  may  be  made  to  touch  one 
limb  while  e  touches  the  other.  Read  off  in  the  field  of  view 
how  many  notches  have  been  passed  over  by  each  wire,  and  the 
fractional  part  of  a  revolution,  on  the  divided  heads.  The  sum 
of  the  two  quantities  will  give  the  whole  number  of  revolutions 
and  parts  indicated.  If  the  distance  to  be  measured  is  small,  it 

C 


34  PRACTICAL   ASTRONOMY. 

will  only  be  necessary  to  move  one  of  the  lines  while  the  other 
remains  at  zero. 

The  micrometers  made  at  Munich  differ  from  the  preceding 
in  having  but  a  single  graduated  head,  and  only  one  of  the 
threads  is  designed  to  be  moved  in  observations.  The  other 
thread  is  only  allowed  a  slight  motion  to  adjust  for  index  error. 

To  find  the  value  of  one  revolution  of  the  screw. 

(40.)  First  Method.  —  Find  how  many  revolutions  of  the 
screw  will  exactly  measure  the  vertical  diameter  of  the  sun 
when  his  altitude  is  considerable,  allowance  being  made  for  the 
difference  of  refraction  of  the  upper  and  lower  limbs.  The 
whole  diameter,  as  given  by  the  Nautical  Almanac,  reduced  to 
seconds,  and  divided  by  the  number  of  revolutions  and  decimal 
parts  observed  on  the  head  of  the  screw,  will  give  the  value  of 
a  single  revolution. 

Example.  When  the  sun's  altitude  was  40°  307,  his  vertical 
diameter  was  measured  by  40.98  revolutions  of  the  micrometer 
screw.     The  sun's  diameter,  according  to  the  Nautical  Almanac, 
was  31/  31/7.2.     The  refraction  of  the  lower  limb  was  computed 
to  be  l/x.2  greater  than  that  of  the  upper  limb ;  hence  the  ap- 
parent vertical  diameter  of  the  sun  was  1890".     Therefore,  the 
value  of  one  revolution  of  the  screw  was 
1890 
40798= 

(41.)  Second  Method. — Separate  the  two  lines  by  any  con- 
venient number  of  revolutions,  and  observe  the  time  required 
by  an  equatorial  star  to  pass  from  one  line  to  the  other.  We 
thus  obtain  the  interval  between  the  two  lines  in  seconds  of 
time,  which,  divided  by  the  number  of  revolutions,  gives  the 
value  of  one  revolution  of  the  screw.  This  result  will  be  the 
more  reliable,  the  greater  the  distance  between  the  lines,  because 
the  unavoidable  error  in  estimating  a  fraction  of  a  second  of 
time  is  reduced  by  a  larger  divisor  ;  and  the  observation  should 
be  repeated  until  a  satisfactory  result  is  obtained.  The  same 
result  may  be  obtained  from  a  star  situated  out  of  the  equator, 
by  reducing  the  observed  interval,  in  the  manner  described  in 
Art.  72.  This  method  will  be  illustrated  by  an  example  with 
the  transit  instrument,  Art.  73. 


THE    MICROMETER. 


POSITION    MICROMETER. 

(42.)  It  is  often  required  to  measure  the  angle  which  the 
line  joining  the  centres  of  two  stars  makes  with  the  meridian. 
This  is  effected  by  means  of  rack-work  and  a  screw,  carrying 
the  spider-line  micrometer  round  in  a  circle,  at  right  angles  to 
the  axis  of  the  telescope, 
and  the  motion  is  meas- 
ured by  means  of  a  grad- 
uated circle. 

The  annexed  figure  rep- 
resents the  spider-line  mi- 
crometer attached  to  the 
end  of  a  telescope,  and  hav- 
ing, besides,  a  graduated 
circle,  AA,  with  a  milled 
head,  S,  acting  upon  a  con- 
cealed rack,  by  which  the 
micrometer,  BB,  may  be 
made  to  revolve  entirely 
round  the  axis  of  the  tele- 
scope. This  motion  is  measured  upon  the  circle  AA  by  means 
of  the  fixed  index  C. 

In  order  to  measure  the  angle  of  position  of  two  stars,  point 
the  telescope  upon  one  of  the  stars,  and  turn  the  micrometer 
until  the  line  /  (see  figure,  page  33)  is  made  to  bisect  the  star 
during  the  whole  time  of  its  crossing  the  field  of  view ;  this 
line  will  then  be  parallel  with  the  equator.  Let  the  index  of 
the  position  circle  now  be  put  to  zero.  Then  revolve  the  mi- 
crometer until  the  line  I  can  be  made  to  bisect  both  stars  at  the 
same  time,  and  note  the  reading  of  the  position  circle.  This 
will  be  the  angle  of  position  measured  from  a  parallel  of  decli- 
nation, as  practiced  by  Sir  William  Herschel. 

(43.)  An  equatorial,  furnished  with  a  micrometer,  affords  the 
most  convenient  means  of  determining  the  position  of  a  comet, 
by  comparing  it  with  some  neighboring  star.  The  method  of 
observation  is  as  follows  :  The  equatorial  having  been  previous- 
ly adjusted,  point  the  telescope  upon  some  convenient  star,  and 
make  the  wire  /of  the  micrometer  (see  figure,  page  33)  parallel 


36  PRACTICAL   ASTRONOMY. 

to  the  equator,  which  is  known  to  be  the  case  when  a  star  will 
travel  along  the  wire  during  its  passage  through  the  field  of  the 
telescope.  Then  turn  the  circle  90°,  and  the  line  /  will  be  per- 
pendicular to  the  equator.  Now  point  the  telescope  upon  the 
comet,  and,  having  clamped  both  the  hour  and  declination  cir- 
cles very  firmly,  note  by  the  clock  the  time  when  the  comet 
passes  over  the  wire  /,  bisecting  it  at  the  same  time  by  the  wire 
e.  Wait  till  the  star  of  comparison  passes  over  the  field,  note 
its  transit  over  the  wire  /,  and  bisect  it  in  declination  with  the 
wire  d,  by  turning  the  head,  g,  of  the  micrometer  screw.  Then 
the  difference  of  the  times  of  observation  gives  the  difference  of 
right  ascension  between  the  comet  and  star,  and  the  difference 
of  declination  is  taken  from  the  micrometer.  If  the  place  of 
the  star  of  comparison  is  not  already  known,  it  must  be  after- 
ward observed  by  meridian  instruments,  and  then  the  place  of 
the  comet  is  deduced  with  the  greatest  accuracy. 
(44.)  Method  of  illuminating  the  Lines. 
Hitherto  it  has  been  presumed  that  the  spider  lines  in  the 
micrometer  will  be  visible  in  all  celestial  observations,  made  by 
night  as  well  as  by  day  ;  whereas,  in  all  nocturnal  observations, 
artificial  light  is  required  to  render  the  lines  of  the  micrometer 
visible.  This  light  is  supplied  by  opening  a  circular  hole  in  the 
side  of  the  main  tube,  before  which  a  lamp  is  suspended,  and 
placing  an  oval  ring  of  gilt  metal,  deadened  so  as  to  reflect  a 
mitigated  light  up  the  telescope,  at  a  proper  angle  of  inclination 
within  the  tube.  In  order  to  regulate  the  quantity  of  light,  so 
as  not  to  conceal  faint  stars  from  view,  variable  diaphragms  may 
be  interposed,  or  darkening  glasses  of  different  shades  of  color. 

By  means  of  a  small  lamp, 
fitted  to  an  aperture  in  the  tube 
next  to  the  eye-piece,  the  wires 
may  be  illuminated  while  the 
field  remains  dark,  thus  enabling 
the  observer  to  have  bright  lines 
and  a  dark  field,  or  a  bright  field 
and  dark  lines,  at  pleasure. 

The  annexed  diagram  repre- 
sents the  illuminated  lines  as 
they  are  usually  arranged  in  a 


COMET    SEEKER. 


37 


small  transit  instrument,  with  a  star  just  going  off  on  the  left 
side,  and  the  planet  Venus  approaching  to  the  first  wire. 

(45.)  Sir  William  Herschel  was  accustomed  to  distinguish 
angles  of  position  by  the  terms  north,  following,  south,  preced- 
ing ;  which  words  were  designated  by  the  initials  n,  /,  s,  p  ; 
and  he  measured  angles  only  up  to  90° ;  beginning  at  the  pre- 
ceding and  following  points,  and  reading  each  way,  north  and 
south,  up  to  90°.  It  is  now  more  common  to  measure  angles 
of  position  from  the  north  point  by  the  east,  round  to  360°. 
The  following  diagram  shows  both  forms,  as  used  in  the  re- 
versed field  of  a  telescope. 


COMET-SEEKER. 

(46.)  The  comet-seeker  is  a  telescope  having  an  object-glass 
of  large  aperture  and  short  focal  length,  with  which  low  mag- 
nifying powers  are  used,  that  it  may  have  a  large  field  of  view, 
and  collect  the  greatest  possible  amount  of  light.  The  figure 
on  the  next  page  represents  such  an  instrument  mounted  equa- 
torially.  It  rests  upon  a  tripod,  with  foot  screws,  H,  H,  H,  for 
leveling.  From  the  tripod  rises  a  vertical  shaft,  whose  upper 
extremity  is  enlarged  for  the  support  of  an  axis,  P,  parallel  to 
the  axis  of  the  earth.  To  this  is  attached  an  hour  circle,  Gr, 


38 


PRACTICAL   ASTRONOMY. 


graduated  to  hours,  minutes,  and  seconds ;  and  upon  its  edge 

are  cut  threads,  to  receive  an 
endless  screw,  M,  which  com- 
municates a  slow  motion 
about  the  axis.  At  right  an- 
gles to  the  polar  axis  is  the 
declination  axis,  with  its  cir- 
cle, E,  divided  into  degrees 
and  minutes,  and  having  also 
a  tangent  screw  for  slow  mo- 
tion. The  telescope  tube,  T, 
is  of  the  ordinary  construc- 
tion, and  is  accurately  coun- 
terpoised by  the  weight  "W. 
The  shaft  has  a  level,  L,  for 
adjusting  it  to  a  vertical  po- 
sition by  means  of  the  foot 
screws,  and  a  tangent  screw, 
bd,  gives  a  slow  motion  in 
azimuth. 

(47.)  The  adjustments  of 
this  instrument  are  the  same 
as  those  of  a  large  equatorial ; 
but  inasmuch  as  it  is  design- 

V1 Vj^ *¥  ed  merely  to  scour  the  heav- 

ens  in  search  of  comets,  and 

not  for  accurate  observations  of  position^  no  great  precision  is 

usually  aimed  at  in  the  adjustments. 


T 


ir 

TT 


CHAPTER  II. 

THE  TRANSIT  INSTRUMENT. 

(48.)  THE  transit  is  a  meridional  instrument,  employed,  in 
connection  with  a  clock  or  chronometer,  for  observing  the  pas- 
sage of  celestial  objects  across  the  meridian,  either  for  obtain- 
ing correct  time  or  determining  their  right  ascensions. 

(49.)  The  figure  on  the  next  page  represents  a  portable  transit 
instrument.  The  telescope  tube,  AA,  is  supported  upon  an  axis, 
BB,  placed  at  right  angles  to  the  direction  of  the  telescope. 
This  axis  terminates  in  two  cylindrical  pivots  which  rest  in  Y's, 
which  are  strongly  united  to  the  two  uprights  CC.  The  stand 
CDC,  carrying  the  Y's,  is  made  of  cast-iron,  and  should  be  made 
of  a  single  piece,  in  order  to  secure  a  steady  and  permanent  po- 
sition. At  the  left  end  of  the  axis  there  is  a  screw,  E ,  by  which 
the  Y  of  that  extremity  may  be  raised  or  lowered  a  little,  in 
order  that  the  axis  may  be  made  perfectly  horizontal.  At  the 
right  end  of  the  axis  is  a  screw,  F,  by  which  the  Y  of  that  ex- 
tremity may  be  moved  backward  or  forward,  in  order  to  enable 
us  to  bring  the  telescope  into  the  plane  of  the  meridian.  Near 
the  right  end  of  the  axis  is  fixed  a  circle,  G-,  which  turns  with 
the  axis,  while  the  vernier,  H,  remains  stationary  in  a  horizon- 
tal position,  and  shows  the  altitude  to  which  the  telescope  is 
elevated.  The  vernier  is  set  horizontal  by  means  of  an  attached 
spirit  level,  I.  The  level  KK  rides  on  the  pivots  of  the  axis. 
There  is  a  pin  at  each  end,  which  drops  into  a  fork  at  L,  to  hold 
the  level  safely  and  upright.  At  the  left  end  is  the  adjustment 
for  setting  the  level  tube  parallel  with  the  axis.  At  the  other 
end  is  an  adjustment  for  raising  or  depressing  the  extremity  of 
the  level. 

(50.)  Near  the  eye  end,  and  in  the  principal  focus  of  the 
telescope,  is  placed  the  diaphragm  or  wire  plate,  carrying  five 
or  seven  vertical  wires,  and  two  horizontal  ones,  between  which 
the  star  is  observed.  The  central  vertical  wire  ought  to  be 
fixed  in  the  optical  axis  of  the  telescope,  and  perpendicular 


40 


PRACTICAL   ASTRONOMY. 


THE    TRANSIT   INSTRUMENT.  41 

with  respect  to  the  pivots  of  the  axis.  These  wires  are  rendered 
visible  in  the  day  time  by  the  diffuse  light  of  day,  which  pene- 
trates the  tube  of  the  telescope  ;  but  at  night,  artificial  illu- 
mination is  required.  This  illumination  is  effected  by  piercing 
one  of  the  pivots,  and  admitting  the  light  of  a  lamp,  M,  fixed 
on  the  top  of  one  of  the  standards.  This  light  is  directed  to 
the  wires  by  a  reflector,  placed  diagonally  at  the  junction  of 
the  axis  and  telescope  ;  the  reflector  having  a  large  hole  in  its 
centre,  so  as  not  to  interfere  with  the  rays  passing  down  the 
telescope  from  the  object.  By  inclining  the  opening  of  the 
lantern,  more  or  less  light  may  be  admitted  to  the  telescope,  to 
accommodate  faint  objects,  which  might  be  entirely  eclipsed 
by  a  bright  light.  The  telescope  is  furnished  with  a  diagonal 
eye-piece,  W,  by  which  stars  near  the  zenith  may  be  observed 
without  inconvenience.  The  head  of  the  micrometer  screw  is 
shown  at  P. 

(51.)  A  transit  instrument  for  a  large  observatory  differs  from 
the  preceding  chiefly  in  being  of  larger  dimensions,  and  resting 
upon  stone  piers  instead  of  a  movable  frame.  The  figure  on 
the  next  page  represents  such  an  instrument,  as  made  by  Ertel 
and  Son,  of  Munich.  PP  are  the  stone  piers,  which  rest  upon 
foundations  sunk  deep  in  the  earth.  The  axis  of  the  telescope 
is  made  strong,  so  as  to  resist  flexure,  and  was  cast  in  a  single 
piece,  the  middle  part  of  it  being  in  the  form  of  a  cube.  The 
telescope  tube  is  composed  of  two  conical  frustums,  which 
are  fastened  by  screws  to  the  cubical  part  of  the  axis.  The 
weight  of  a  seven-feet  transit  is  about  200  pounds.  In  order 
that  the  pivots  may  be  relieved  from  a  portion  of  this  weight, 
there  is  raised  upon  the  top  of  each  pier  a  brass  pillar,  E,  about 
18  inches  high.  On  the  top  of  the  pillar  there  is  a  lever,  H, 
from  one  end  of  which  hangs  a  strong  brass  hook,  X,  support- 
ing two  friction  rollers  under  the  ends  of  the  great  axis.  A 
counterpoise,  W,  sliding  on  the  other  end  of  the  lever,  may  be 
made  to  support  as  much  of  the  weight  of  the  instrument  as  is 
desired. 

(52.)  Both  the  pivots  of  the  axis  are  perforated  to  admit  the 
light  of  a  lamp,  on  an  elliptic-ring  reflector  placed  inside  the 
square  part  of  the  axis.  To  moderate  the  light  of  the  lamp, 
L,  there  is  a  green  glass  wedge,  movable  up  and  down  between 


42 


PRACTICAL    ASTRONOMY. 


THE    TRANSIT    INSTRUMENT. 


43 


the  lamp  and  pivot.  The  thinnest  part  of  the  wedge  transmits 
nearly  all  the  light  of  the  lamp,  while  the  thickest  part  trans- 
mits only  as  much  light  as  is  barely  sufficient  to  show  the 
spider  lines.  On  each  side  of  the  eye  end  of  the  telescope  is  a 
finding  circle,  D,  with  a  level,  by  which  the  telescope  can  be  set 
to  any  zenith  distance. 

There  are  also  in  the  eye-tube,  at  the  back  of  the  plate  car- 
rying the  spider  lines,  two  oblong  openings  covered  with  glass. 
A  lighted  taper  held  near  either  of  these  apertures  illumines  the 
lines  without  enlightening  the  field  of  view,  by  which  means 
very  small  stars  can  be  observed,  which  could  not  be  seen  by 
the  ordinary  illumination.  Z  represents  the  handle  of  the  screw 
which  fastens  the  clamp  arm  to  the  axis  of  the  instrument ; '  B 
and  C  are  handles,  by  which 
a  slow  motion  in  altitude  is 
given  to  the  telescope ;  00 
is  the  eye-piece,  with  spider 
lines,  micrometer,  etc. 

(53.)  It  is  frequently  neces- 
sary to  reverse  the  axis  of  the 
transit  instrument,  and  large 
telescopes  need  some  special 
contrivance  by  which  this 
may  be  readily  accomplished. 
The  most  convenient  appara- 
tus for  this  purpose  is  a  revers- 
ing stand,  represented  in  the 
annexed  figure.  The  stand, 
being  on  rollers,  is  brought 
under  the  axis  of  the  transit, 
when,  by  turning  the  handle 
K,  the  rod  MN  is  elevated  by 
means  of  a  screw,  and  the 
instrument  lifted  from  the 
piers  by  means  of  the  forks 
RR.  The  stand  is  then  rolled 
away  from  between  the  piers, 
and  the  instrument  turned 
half  round.  The  stand  is 


44  PRACTICAL   ASTRONOMY. 

again  rolled  between  the  piers,  and  the  axis  returned  to  its 
place. 

ADJUSTMENTS. 

(54.)  When  the  instrument  is  set  up,  it  should  be  so  placed 
that  the  telescope,  if  turned  down  to  the  horizon,  may  point 
north  and  south  as  near  as  can  possibly  be  ascertained.  This, 
of  course,  can  only  be  done  approximately,  as  the  meridian  can 
not  be  accurately  determined  until  the  other  adjustments  have 
been  completed. 

Distinctness  of  vision  and  parallax. 

(55.)  The  system  of  wires  or  spider  lines  should  be  in  the 
common  focus  of  the  object-glass  and  eye-glass.  In  order  to 
place  the  lines  in  the  focus  of  the  eye-glass,  push  in  or  draw 
out  the  eye-tube  until  they  are  seen  with  perfect  distinctness. 
Now,  if  the  wires  are  not  in  the  focus  of  the  object-glass  when 
the  telescope  is  directed  toward  a  distant  mark,  if  the  eye  be 
moved  a  little  to  the  right  or  left,  the  mark  will  appear  to  move 
with  reference  to  the  lines.  When  this  is  the  case,  the  object- 
glass  or  the  wires  must  be  moved  in  the  tube  until  the  parallax 
is  corrected,  after  which  they  must  be  secured  firmly  to  their 
places.  After  the  transit  has  been  placed  in  the  meridian,  and 
the  wires  adjusted  as  described  hereafter,  let  a  star  run  along 
the  horizontal  wire,  and  if  it  does  not  remain  perfectly  bisected 
while  the  eye  is  moved  up  and  down,  the  adjustment  for  par- 
allax is  not  complete. 

Horizontality  of  the  axis. 

(56.)  The  axis  on  which  the  telescope  turns  must  next  be 
made  horizontal.  This  is  effected  by  means  of  the  level.  The 
level  is  a  glass  tube,  apparently  cylindrical,  but  in  reality  a  por- 
tion of  &  ring  of  very  large  radius,  nearly  filled  with  spirit  of 
wine  or  sulphuric  ether.  The  convex  side  being  placed  upward, 
the  bubble  will  occupy  the  higher  part, 
as  ab  ;  and  if  either  end  of  the  level 
be  elevated,  the  bubble  will  move  in 
that  direction.  If,  then,  a  divided  scale  be  attached  to  the  level, 
the  motion  of  the  bubble  will  measure  the  elevation  of  the  end 
of  the  level.  The  figure  on  the  opposite  page  shows  the  com- 
mon form  of  level,  which  should  be  made  of  such  dimensions, 


THE    TRANSIT   INSTRUMENT. 


45 


that  the  legs  may  extend  from  one  pivot  of  the  transit  instru- 
ment to  the  other. 


D 


Fig.  1  represents  a  front  view  of  the  level,  Figs.  2  and  3  rep- 


rig.  2. 


/~\J 


resent  end  views  of  the  legs.  The  level 
consists  of  a  glass  tube,  GGr,  which  lies 
in  a  semi-cylinder  of  brass,  BB,  and  is 
secured  to  it  by  two  thin  brass  straps, 
DD.  The  cylinder  is  connected  with 
one  leg  of  the  level  by  the  two  screws, 
SS',  seen  in  Fig.  2,  and  with  the  other 


rig.  s. 


leg  by  the  screws  TTX,  seen  in  Fig.  3.  The  screws  SS' 
serve  to  move  the  cylinder  BB  in  a  horizontal  direction;  the 
screws  TT'  serve  to  move  it  in  a  vertical  direction.  Each  foot 
of  the  level  has  two  planes,  inclined  at  an  angle  of  60  to  90  de- 
grees, which  are  designed  to  rest  upon  the  pivots  of  the  transit. 

(57.)  The  level  should  be  so  adjusted  that  its  axis  may  be 
parallel  with  the  axis  of  the  transit.  For  this  purpose,  place  the 
level  upon  the  pivots  of  the  axis,  and  bring  the  air-bubble  to  the 
centre  of  the  glass  tube  by  turning  the  screw  which  raises  or 
fowers  the  end  of  the  axis.  Then  reverse  the  level  so  that  the 
end  which  before  rested  on  the  right  pivot  may  rest  on  the  left. 
If  the  bubble  settles  in  the  same  position  as  before,  we  may  con- 
clude that  the  axis  of  the  transit  is  horizontal  ;  but  if  the  bub- 
ble moves  from  its  former  position,  the  amount  of  this  motion 
will  be  equal  to  twice  the  inclination  of  the  axis  to  the  horizon. 
Turn  the  screw  at  the  end  of  the  axis  so  as  to  move  the  bubble 
over  half  this  distance  ;  then  loosen  one  of  the  screws,  TT',  Fig. 
3,  and  tighten  the  other,  until  the  bubble  is  brought  back  to  the 
middle  of  the  tube. 

Since  it  is  difficult  to  make  this  adjustment  perfect  at  a  sin- 


46  PRACTICAL   ASTRONOMY. 

gle  trial,  we  must  repeat  the  same  series  of  operations  until  the 
bubble  occupies  the  same  place  in  both  the  direct  and  reversed 
positions  of  the  level.  When  this  is  accomplished,  the  axis  of 
the  level  will  be  in  a  plane  which  is  parallel  to  the  axis  of  the 
transit,  but  the  two  axes  will  not  necessarily  be  parallel  with 
each  other.  To  determine  whether  such  is  the  case,  revolve  the 
level  slightly  upon  the  axis  of  the  transit,  the  feet  of  the  level 
remaining  all  the  while  in  contact  with  the  pivots.  If  the  bub- 
ble changes  its  place,  the  axis  of  the  level  must  be  inclined  to 
the  axis  of  the  transit,  and  we  must  turn  the  screws  SSX,  Fig. 
2,  either  forward  or  backward,  until  a  slight  rotation  of  the  level 
about  the  axis  of  the  transit  causes  no  sensible  change  in  the  po- 
sition of  the  bubble.  When  this  second  correction  is  completed, 
the  former  must  be  verified  anew. 

(58.)  To  discover  whether  the  level  is  well  made,  place  it 
upon  a  rule,  having  at  one  end  two  points,  which  enter  two  cor- 
responding cavities  upon  an  iron  bar,  while  at  the  other  end  of 
the  rule  is  a  delicate  micrometer  screw,  pressing  firmly  against 
a  cavity  in  the  iron  bar.  The  whole  must  be  placed  upon  a 
very  firm  support.  Then,  upon  turning  the  micrometer  screw 
so  as  to  change  the  inclination  of  the  level  to  the  horizon,  it  will 
be  easily  seen  whether  equal  parts  of  a  rotation  of  the  screw 
correspond  to  equal  movements  of  the  bubble  along  the  glass 
tube.  When  this  is  the  case,  the  level  is  good.  By  means  of 
this  arrangement  we  may  easily  determine  the  value  of  one  di- 
vision of  the  level,  expressed  in  seconds  of  arc.  Measure  the 
distance  of  the  cavity  in  which  the  micrometer  screw  rests  from 
the  middle  of  the  line  connecting  the  two  other  cavities  in  the 
iron  bar,  and  represent  this  distance,  expressed  in  inches  and 
parts  of  an  inch,  by  d.  Count  the  number  of  threads  contained 
in"  an  inch  upon  the  screw,  from  which  we  can  determine  the 
distance  between  two  threads,  expressed  in  parts  of  an  inch. 
Represent  this  distance  by  b.  Then,  it  is  plain  that  the  incli- 
nation of  the  level  to  the  horizon  will  be  changed  in  one  revo- 
lution of  the  screw  by  an  angle  equal  to  --— — ^-7>.  If,  therefore, 

d>  Sill*  _L 

we  know  how  many  divisions  of  the  level  correspond  to  one  rev- 
olution of  the  screw,  we  may  determine  the  value  of  one  division 
of  the  level,  expressed  in  parts  of  a  second. 


THE    TRANSIT   INSTRUMENT.  47 

(59.)  We  may  also  determine  the  value  of  one  division  of  the 
level  in  the  following  manner : 

Fix  the  level  to  the  tube  of  a  telescope  connected  with  a  vert- 
ical divided  circle  reading  to  seconds.  Move  the  telescope  by 
means  of  the  tangent  screw,  so  as  to  carry  the  bubble  success- 
ively to  one  side  and  the  other  of  the  level,  and  read  off  the  cir- 
cle in  the  two  positions.  The  difference  of  these  readings  in 
seconds,  divided  by  the  number  of  divisions  of  the  level  that  the 
bubble  has  moved,  will  give  the  value  of  one  division.  Delicate 
levels  are  generally  designed  to  be  divided  in  such  a  manner 
that  one  division  shall  represent  one  second  of  arc.  At  one  end 
of  the  level-tube  are  small  screws,  by  which  that  end  may  be 
elevated  or  depressed,  so  as  to  bring  the  bubble  into  the  middle 
of  the  tube  when  the  level  is  placed  on  a  horizontal  surface. 

Perpendicularity  of  the  wires. 

(60.)  It  is  desirable  that  the  central  or  middle  wire  should 
be  truly  vertical,  as  we  may  then  observe  the  transit  of  a  star 
on  any  part  of  it  as  well  as  the  centre.  For  this  purpose,  direct 
the  telescope  upon  a  small,  well-defined,  and  distant  object.  If, 
on  moving  the  telescope  in  altitude,  this  mark  is  perfectly  bi- 
sected by  the  central  wire  from  top  to  bottom,  the  wire  is  per- 
pendicular to  the  horizontal  axis.  If  not,  the  ring  or  tube  con- 
taining the  wires  must  be  turned  round  until  the  mark  is  bi- 
sected by  every  part  of  the  wire.  The  other  vertical  wires  are 
placed  by  the  maker  as  nearly  as  possible  equidistant  from  each 
other,  and  parallel  to  the  middle  one ;  therefore,  when  the  mid- 
dle one  is  adjusted,  the  others  are  also  adjusted.  The  trans- 
verse wires  are  also  placed  at  right  angles  to  the  vertical  mid- 
dle wire. 

Collimation. 

(61.)  The  optical  axis  of  a  lens  is  the  line  which  joins  the 
centres  of  the  spherical  surfaces  by  which  the  lens  is  bounded. 
When  a  telescope  is  properly  constructed,  the  axes  of  the  object- 
glass  and 'eye-glass  must  lie  in  the  straight  line  which  joins  the 
centres  of  the  object-glass  and  eye-glass.  This  straight  line  is 
called  the  optical  axis  of  the  telescope. 

The  principal  line  of  sight,  or  the  line  of  collimation,  is  de- 
termined by  the  direction  of  the  ray  of  light  which  passes 
through  the  centre  of  the  object-glass,  and  touches  the  middle 


48  PRACTICAL   ASTRONOMY. 

vertical  thread  midway  between  the  two  horizontal  threads.  In 
the  rotation  of  the  telescope  about  its  axis,  the  line  of  collima- 
tion  should  describe  a  plane  perpendicular  to  this  axis.  To  de- 
termine whether  such  is  actually  the  case,  direct  the  telescope 
to  some  small,  well-defined,  and  distant  object,  and  bisect  it  with 
the  middle  vertical  wire.  Then  lift  the  telescope  very  carefully 
from  its  supports,  and  replace  it  with  the  axis  reversed.  Point 
the  telescope  again  to  the  same  object,  and  if  it  be  still  bisected, 
the  collimation  adjustment  is  correct ;  if  not,  move  the  wires 
one  half  the  error  by  turning  the  small  screws  which  hold  the 
diaphragm,  near  the  eye  end  of  the  telescope.  But  as  half  the 
deviation  may  not  be  correctly  estimated  in  moving  the  wires, 
it  becomes  necessary  to  verify  the  adjustment  by  moving  the 
telescope  the  other  half,  which  is  done  by  turning  the  screw  F 
(see  figure,  page  40).  Having  again  bisected  the  object,  reverse 
the  axis  as  before,  and,  if  half  the  error  was  correctly  estimated, 
the  object  will  be  bisected  when  the  telescope  is  directed  to  it. 
If  this  is  found  not  to  be  the  case,  half  the  remaining  error  must 
be  corrected  as  before,  and  these  operations  must  be  continued 
until  the  object  is  found  to  be  bisected  in  both  positions  of  the 
axis.  The  adjustment  will  then  be  complete. 

POSITION    IN    THE    MERIDIAN. 

(62.)  This  adjustment  is  effected  with  the  assistance  of  a 
clock,  which$  for  convenience,  should  be  regulated  to  sidereal 
time,  so  that  the  time  of  each  star's  passing  the  meridian  will 
be  indicated  by  its  right  ascension. 

By  the  pole  star. 

(63.)  Direct  the  telescope  to  the  pole  star  at  the  instant  of 
its  crossing  the  meridian,  as  near  as  the  time  can  be  ascertained. 
The  transit  will  then  be  nearly  in  the  plane  of  the  meridian. 
Having  leveled  the  axis,  turn  the  telescope  to  a  star  about  to 
cross  the  meridian,  near  the  zenith.  Since  every  vertical  circle 
intersects  the  meridian  at  the  zenith,  a  zenith  star  will  cross  the 
field  of  the  telescope  at  the  same  time,  whether  the  plane  of  the 
transit  coincide  with  the  meridian  or  not.  At  the  moment  the 
star  crosses  the  central  wire,  set  the  clock  to  its  right  ascension, 
as  given  by  the  catalogue,  and  the  clock  will  henceforth  indicate 
nearly  sidereal  time.  The  approximate  times  of  the  upper  and 


THE    TRANSIT   INSTRUMENT.  49 

lower  culmination  of  the  pole  star  are  then  known.  Observe 
the  pole  star  at  one  of  its  culminations,  following  its  motion  un- 
til the  clock  indicates  its  right  ascension,  or  its  right  ascension 
plus  12  hours.  Move  the  whole  frame  of  the  transit,  so  that 
the  central  wire  shall  coincide  nearly  with  the  star,  and  com- 
plete the  adjustment  by  means  of  the  azimuth  screw.  The  cen- 
tral wire  will  now  coincide  almost  precisely  with  the  meridian 
of  the  place. 

(64.)  The  axis  being  supposed  perfectly  horizontal,  if  the  mid- 
dle wire  of  the  telescope  is  exactly  in  the  meridian,  it  will  bi- 
sect the  circle  which  the  pole  star  describes,  in  24  sidereal  hours, 
round  the  polar  point.  If,  then,  the  interval  between  the  upper 
and  lower  culminations  is  exactly  equal  to  the  interval  between 
the  lower  and  upper,  the  adjustment  is  complete.  But  if  the 
time  elapsed  while  the  star  is  traversing  the  eastern  semicircle 
is  greater  than  that  of  traversing  the  western,  the  plane  in 
which  the  telescope  moves  is  westward  of  the  true  meridian  on 
the  north  horizon ;  and  vice  versa,  if  the  western  interval  is 
greatest.  This  error  must  be  corrected  by  turning  the  screw  F 
(page  40).  The  adjustment  must  then  be  verified  by  further 
observations,  until,  by  continued  approximations,  the  instrument 
is  fixed  correctly  in  the  meridian. 

By  a  pair  of  circumpolar  stars. 

(65.)  Take  two  well-known  circumpolar  stars,  the  nearer  the 
pole  the  better,  differing  about  twelve  hours  in  right  ascension, 
and  observe  one  above  and  the  other  below  the  pole.  Now  it  is 
evident  that  any  deviation  of  the  instrument  from  the  meridian 
will  produce  contrary  effects  upon  the  observed  times  of  transit. 
Hence  the  time  which  elapses  between  the  two  observations 
will  differ  from  the  time  which  should  elapse  according  to  the 
catalogue,  by  the  sum  of  the  effects  of  the  deviation  upon  the 
two  stars.  The  stars  51  Cephei  and  6  Ursse  Minoris  are  well 
suited  to  this  purpose.  The  right  ascension  of  the  former  on  the 
1st  of  January,  1855,  was  6h;  31m.  26s. ;  of  the  latter,  18h.  18m. 
48s.  The  difference  between  the  times  at  which  one  should 
make  its  upper  and  the  other  its  lower  transit  is  12m.  38s.  If 
the  observed  interval  differs  from  this,  the  error  must  be  correct- 
ed by  the  azimuth  screw,  and  the  observations  repeated  until 
the  adjustment  is  perfect. 

D 


50  PRACTICAL    ASTRONOMY. 

By  the  pole  star,  combined  with  any  star  distant  from  the 
pole. 

(66.)  If  the  transit  moves  in  the  plane  of  the  meridian,  the 
error  of  the  clock,  as  determined  by  the  culmination  of  the  pole 
star,  will  be  exactly  the  same  as  from  any  other  star  situated, 
for  example,  near  the  equator.  But  if  the  transit  describes  a 
vertical  circle  which  differs  from  the  meridian,  the  pole  star  will 
be  longer  in  crossing  from  the  transit  plane  to  the  true  meridian 
than  the  equatorial  star.  If,  then,  the  two  stars  do  not  indicate 
the  same  clock  error,  the  azimuth  screw  must  be  moved  until 
the  adjustment  is  perfect. 

By  a  high  and  low  star. 

(67.)  This  method  may  be  practiced  in  situations  which  do 
not  permit  an  observation  of  the  pole  star.  Choose  two  stars 
differing  but  little  in  right  ascension,  one  of  them  passing  the 
meridian  as  near  as  possible  to  the  zenith,  and  the  other  as  near 
as  convenient  to  the  south  horizon.  Make  the  axis  of  the  transit 
perfectly  horizontal,  so  that  the  transit  shall  describe  a  vertical 
circle.  This  circle  will  coincide  with  the  meridian  at  the  ze- 
nith, however  much  it  may  depart  from  it  at  the  horizon.  A 
star  near  the  zenith  will  pass  the  middle  wire  of  the  telescope  at 
about  the  same  time  as  if  the  transit  was  in  the  meridian  ;  but 
this  will  not  be  the  case  with  a  star  near  the  south  horizon.  If 
the  low  star  passes  the  central  wire  too  early,  the  plane  of  the 
instrument  deviates  to  the  east ;  if  it  passes  too  late,  the  plane 
deviates  to  the  west.  In  either  case  the  error  must  be  corrected 
by  the  azimuth  screw,  until  stars  at  all  altitudes  indicate  the 
same  error  of  the  clock. 

MERIDIAN    MARK. 

(68.)  "When  the  transit  instrument  has  once  been  brought  to 
the  meridian,  a  mark  may  be  placed,  either  to  the  north  or  south, 
for  verification,  in  case  the  instrument  should  at  any  time  be 
disturbed.  It  should  be  placed  at  such  a  distance  as  not  to  be 
affected  by  parallax,  yet  not  too  far  to  be  seen  distinctly.  The 
observatory  at  Edinburgh  has  two  meridian  marks,  one  distant 
about  8000  feet,  and  the  other  18,000  feet  from  the  observatory. 
Each  is  formed  of  a  piece  of  copper,  having  an  aperture  of  the 
figure  of  an  isosceles  triangle,  the  base,  parallel  to  the  horizon, 


THE    TRANSIT    INSTRUMENT.  51 

subtending  an  angle  of  6"  of  space,  and  the  height  being  double 
the  base.  Through  this  aperture  is  seen  a  piece  of  metal  paint- 
ed white. 

THE     CLOCK ITS    RATE    AND    ERROR. 

(69.)  A  clock  designed  to  be  used  for  astronomical  purposes 
should  be  of  the  best  workmanship.  The  pendulum  should  be 
compensated,  so  as  to  be  free  from  the  effects  of  heat  and  cold. 
The  two  forms  chiefly  used  for  astronomical  purposes  are  the 
mercurial  and  the  gridiron.  The  former  is  generally  used  in 
England,  the  latter  in  France  and  Germany. 

It  is  most  convenient  to  have  the  clock  regulated  to  sidereal 
time,  and  it  is  desirable  that  it  should  keep  exact  pace  with  the 
stars,  so  as  always  to  indicate  the  exact  right  ascension  of  the 
star  then  passing  the  meridian.  But  every  clock  has  both  an 
error  and  rate.  The  error  of  the  clock  at  any  time  is  its  dif- 
ference from  true  sidereal  time.  The  rate  of  the  clock  is  the 
change  of  its  error  in  24  hours.  Thus,  on  the  8th  of  January, 
1851,  Aldebaran  was  observed  to  pass  the  meridian  of  Green- 
wich at  4h.  26m.  52.02s.  The  true  right  ascension  of  the  star 
was  4h.  27m.  22.86s. ;  hence  the  clock  was  slow  30.84s.  Again, 
on  the  9th  of  January,  the  same  star  passed  the  meridian  at  4h. 
26m.  51.22s.,  and  the  clock  was  slow  31.64s.  Hence  the  clock 
lost  0.80s.  per  day.  In  other  words,  the  error  of  the  clock,  Jan- 
uary 9th,  was  —31.64s.,  and  its  daily  rate  —0.80s. 

(70.)  The  preceding  error  and  rate  do  not  necessarily  imply 
any  imperfection  of  the  clock.  The  error  and  rate  of  a  perfect 
clock  may  be  of  any  magnitude.  All  which  we  demand  of  a 
clock  is  that  its  rate  be  uniform  from  day  to  day.  Still,  it  is 
convenient  in  practice  that  both  should  be  of  small  amount. 
The  rate  of  the  clock  may  be  corrected  by  lowering  the  bob  of 
the  pendulum,  if  the  clock  runs  too  fast,  or  raising  it  when  the 
clock  runs  too  slow.  For  this  purpose,  the  bob  of  the  pendulum 
is  furnished  with  a  fine  adjusting  screw.  The  clock  may  be 
made  to  indicate  true  sidereal  time  by  setting  it  to  the  right 
ascension  of  any  known  star,  and  starting  the  pendulum  at  the 
moment  when  the  star  crosses  the  middle  transit  wire.  After 
the  rate  has  been  reduced  to  a  small  quantity,  it  is  better  to  let 
the  error  accumulate  than  to  stop  the  clock.  When  the  error 


PRACTICAL    ASTRONOMY. 


amounts  to  a  whole  minute,  the  minute-hand  may  be  moved 
one  division  without  disturbing  the  motion  of  the  pendulum. 
The  transit  clock  at  Greenwich  Observatory  generally  loses 
about  half  a  second  a  day,  and  when  this  error  amounts  to  an 
entire  minute  (which  happens  about  every  three  months),  the 
clock  is  put  forward  one  minute. 

METHOD    OF    OBSERVING    AND    REGISTERING    TRANSITS. 

(71.)  For  a  night  observation,  the  field  of  view  must  be  illu- 
mined by  the  lamp  M  (see  figure,  page  40),  so  that  the  wires 
may  be  distinctly  visible  ;  and  the  telescope  must  be  set  to  the 
proper  altitude  by  means  of  the  attached  circle.  This  circle  is 
sometimes  designed  to  indicate  altitudes  or  zenith  distances, 
and  sometimes  declinations  or  polar  distances.  In  either  case, 
the  zero  of  the  circle  may  require  adjustment.  If  the  circle  in- 
dicates altitudes,  the  index  should  point  to  zero  when  the  bub- 
ble of  the  attached  level  stands  in  the  middle  of  the  tube.  If 
the  circle  indicates  declinations,  the  index  should  point  to  zero 
when  the  telescope  is  directed  toward  an  equatorial  star.  Since 
the  telescope  inverts  the  position  of  objects,  a  star  for  an  upper 
culmination  will  appear  to  enter  the  field  of  view  on  the  west 
side,  and  pass  out  on  the  east ;  but  for  a  lower  culmination,  it 

will  cross  the  field  from  east  to 
west.  The  telescope  contains  five 
or  seven  vertical,  and  two  horizon- 
tal wires,  placed  a  short  distance 
from  each  other.  The  star  should 
be  made  to  cross  the  field  between 
the  two  horizontal  wires,  in  order 
that  the  transits  may  always  be 
observed  on  the  same  part  of  the 
vertical  wires.  It  is  the  business 
of  the  observer  to  note  the  times 
of  the  star's  passage  over  the  several  wires  with  the  utmost  ac- 
curacy ;  and  as  it  will  seldom  happen  that  a  star  will  cross  a 
wire  at  the  exact  instant  of  the  beat  of  the  clock,  he  must  esti- 
mate the  fractions  of  a  second  as  well  as  he  is  able.  This  is 
done  by  comparing  the  distance  of  the  star  from  the  wire  at  the 
beat  preceding  the  transit,  with  its  distance  on  the  other  side  at 


THE    TRANSIT   INSTRUMENT.  53 

the  beat  succeeding  the  transit.  The  clock  should  be  so  placed, 
and  its  face  illumined,  that  the  observer,  seated  at  the  transit, 
can  readily  follow  the  seconds'  hand.  A  little  before  the  star 
is  expected  to  cross  the  first  wire,  the  observer  takes  a  second 
from  the  clock — suppose  5s. — and,  listening  to  the  beats,  goes 
on  silently  counting  6,  7,  8,  9,  etc.,  while  his  eye  is  at  the  tel- 
escope following  the  motion  of  the  star.  If  the  star  crossed  the 
first  wire  between  the  beats  9  and  10,  and  if  the  star  appeared 
as  far  beyond  the  wire  at  the  succeeding  beat  as  it  was  short  of 
it  at  the  preceding  beat,  the  time  of  the  transit  would  be  9.5s. ; 
but  if  the  distances  were  unequal,  it  would  be  9.3s.  or  9.7s.,  etc., 
according  to  its  apparent  distance  from  the  wire.  Having  re- 
corded the  passage  over  the  first  wire,  the  same  observation 
must  be  made  at  each  of  the  other  wires,  and  a  mean  of  the 
whole  taken,  which  will  represent  the  time  of  the  star's  passage 
over  the  mean  or  meridional  wire.  Five  or  seven  wires  are  more 
valuable  than  a  single  one,  since  the  chances  are  that  an  error 
which  may  have  been  committed  at  one  wire  will  be  compen- 
sated by  an  opposite  error  at  another.  Thus  the  mean  result  of 
several  observations  is  deserving  of  more  confidence  than  a  sin- 
gle one.  The  following  is  an  observation  of  Arcturus,  made  at 
Greenwich  Observatory,  November  13th,  1850 : 

First  wire,  14h.  7m.    7.7s. 

Second  wire,  7  21.3 

Third  wire,  7  34.9 

Fourth  wire,  7  48.6 

Fifth  wire,  8        2.1 

Sixth  wire,  8  15.7 

Seventh  wire,  8  29.4 

It  will  be  perceived  that  the  observation  at  the  middle  wire 
differs  0.07s.  from  the  mean  of  the  seven  wires.  If  the  obser- 
vations were  perfect,  and  the  wires  equidistant,  these  two  num- 
bers should  agree  exactly. 

EQUATORIAL    INTERVAL,    OF    THE  .WIRES. 

(72.)  By  comparing  the  transits  of  different  stars,  it  will  be 
seen  that  the  time  occupied  by  a  star  in  traversing  the  interval 
between  the  wires  is  different  on  different  points  of  the  merid- 


>Mean,  14h.  7m.  48.53s. 


54  PRACTICAL    ASTRONOMY. 

ian ;  being  least  at  the  equator,  and  increasing  with  the  distance 
from  that  circle.  The  time  occupied  by  a  star  on  the  equator, 
in  passing  between  any  two  of  the  wires,  is  called  their  equato- 
rial interval ;  and  when  this  interval  is  known,  the  interval  for 
any  parallel  of  declination  may  be  computed. 
Thus,  let  P  be  the  pole  of  the  heavens,  EQ  a 
portion  of  the  equator,  and  BD  a  portion  of  any 
parallel  of  declination;  PBE  and  PDQ  two 
meridians,  but  slightly  inclined  to  each  other. 
3  A  star  at  B  moves  over  the  arc  BD  in  the  same 
time  that  one  at  E  moves  over  EQ,.  But  we 
have 

Geom.,  B.  VI.,  Prop.  13,  Cor.  1, 

arc  EQ  :  arc  BD  : :  CQ :  AD  : :  1 :  cos.  Dec. 
Therefore,  BD  =  EQ  cos.  Dec. 

Now  the  time  in  which  a  star  on  the  parallel  BD  would  move 
over  a  constant  space,  EQ,  must  be,  to  the  time  in  which  an 
equatorial  star  moves  over  the  same,  inversely  as  their  rates  of 
motion,  or  as 

EQ  :  BD  : :  1 :  cos.  Dec. : :  sec.  Dec. :  1. 

(73.)  If,  then,  x  represent  the  equatorial  interval  of  the  wires, 
x  sec.  Dec.  will  be  the  interval  for  any  star.  The  equatorial  in- 
terval may  therefore  be  computed  from  observations  made  upon 
any  star  whose  declination  is  known,  by  multiplying  the  ob- 
served interval  by  the  cosine  of  the  star's  declination.  Thus,  in 
the  preceding  observation  of  Arcturus,  the  difference  between 
each  observation  and  the  mean  of  the  seven  wires  is  as  follows : 


Observed  intervals. 

Equatorial  intervals. 

First  wire, 

40.83s. 

38.376s. 

Second  wire, 

27.23 

25.594 

Third  wire, 

13.63 

12.811 

Fourth  wire, 

0.07 

0.066 

Fifth  wire, 

13.57 

12.755 

Sixth  wire, 

27.17 

25.537 

Seventh  wire, 

40.87 

38.414 

And  if  we  multiply  these  numbers  by  .939908,  the  cosine  of 
the  star's  declination  (19°  57'  50"),  we  shall  obtain  the  equato- 
rial intervals,  as  given  in  the  last  column  above. 

(74.)  The  equatorial  interval  may,  however,  be  obtained  more 


THE    TRANSIT   INSTRUMENT. 


accurately  by  observations  of  a  star  near  the  pole — the  pole  star, 
for  example ;  but  in  this  case  a  slight  modification  of  the  pre- 
ceding rule  becomes  necessary,  for  the  pole  star  does  not  pass 
perpendicularly  from  wire  to  wire,  but  describes  a  considerable 
arc  of  the  small  circle  ABC. 
Now  AD  is  the  sine  of  the 
arc  AB.  In  order,  there- 
fore, to  obtain  the  equate-  - 
rial  intervals  from  the  pole 
star,  we  must  multiply  the 
sine  of  the  observed  interval  by  the  cosine  of  the  declination, 
and  we  shall  obtain  the  sine  of  the  equatorial  interval. 

The  following  observations  of  Polaris,  Dec.  88°  30'  27".0, 
were  made  at  Greenwich  Observatory,  April  26th,  1850 : 


Wires. 

Observations. 

Observed  Intervals. 

Equatorial  Intervals. 

A  

h.      m.         s. 

0  39  48.0 

m.          s. 

-24  43.29 

-38.559 

B  

48     3.0 

-16  28.29 

-25.719 

C 

56  190 

-  8  12.29 

-12.820 

D  

1     4  32.0 

+  0     0.71 

+  0.018 

E  .  . 

12  44.0 

+  8  12.71 

+  12.830 

F  

20  57.0 

+  16  25.71 

+  25.652 

Gr  .    .    . 

29  16.0 

+24  44.71 

+  38.595 

Mean  .... 

1     4  31.29 

The  letters  in  column  first  are  used  to  distinguish  the  wires 
of  the  transit.  The  wires  at  Greenwich  are  designated  by  the 
letters  of  the  alphabet  in  such  a  manner  that,  when  the  illu- 
mined end  of  the  axis  is  east,  the  order  of  the  wires  for  stars 
above  the  pole  is  A,  B,  C,  D,  E,  F,  G- ;  but  when  the  illumined 
end  of  the  axis  is  west,  the  order  is  Gr,  F,  E,  D,  C,  B,  A. 
Column  third  shows  the  difference  between  each  observation 
and  the  mean  of  the  seven  wires.  Column  fourth  shows  the 
equatorial  interval  thence  deduced.  The  fourth  column  is  com- 
puted as  follows : 

24m.  43.29s.  =   6°  10'  49".35  sine =9.0320497  ; 
cos.  Dec.  88°  30'  27".0  =8.4157426; 

38.559s.  =          9'  38."39  sine  =  7.4477923 ; 
and  in  the  same  manner  for  the  other  wires. 

It  will  be  perceived  that  the  middle  wire  differs  slightly  from 
the  mean  of  the  seven  wires,  which  may  be  called  the  mean 


56  PRACTICAL   ASTRONOMY. 

wire.    It  is  customary,  at  Greenwich,  to  reduce  all  observations 
to  the  standard  of  the  mean  wire,  and  not  of  the  middle  wire. 

To  reduce  an  observation  when  all  the  wires  are  not  ob- 
served. 

(75.)  It  may  happen,  through  inadvertence  or  unfavorable 
weather,  that  the  transits  over  only  a  portion  of  the  wires  are 
observed ;  but  such  observations  may  be  reduced  by  means  of 
the  equatorial  intervals  already  determined.  According  to  the 
values  given  above,  if  we  add  38.559s.  to  the  time  of  transit  of 
an  equatorial  star  over  wire  A,  it  will  give  the  time  of  transit 
over  the  mean  wire ;  and,  in  the  same  manner,  the  observation 
of  an  equatorial  star  at  each  wire  may  be  reduced  to  the  mean, 
by  adding  or  subtracting,  as  the  case  may  be,  to  the  time  of 
observation,  the  equatorial  interval  between  that  wire  and  the 
mean  wire.  For  a  star  out  of  the  equator,  these  intervals  must 
each  be  multiplied  by  the  secant  of  the  star's  declination.  Or 
the  following  rule  is  more  convenient  in  practice,  and  evidently 
gives  the  same  result : 

Add  together  the  equatorial  numbers  from  the  table  on  page 
55  for  the  wires  observed,  regard  being  had  to  their  signs  ; 
divide  by  the  number  of  wires,  and  multiply  by  the  secant  of 
the  star's  declination.  The  product  will  be  the  correction  to 
be  applied  to  the  mean  of  the  wires  observed. 

The  corrections  to  transits  of  an  equatorial  star  over  wires  A, 
B,  C,  D,  E,  F,  a,  for  1851,  at  Greenwich,  were  +  41.443s. ; 
+  27.646s. ;  +  13.816s. ;  -0.002s.;  -13.811s.;  -27.654s.; 
—41.438s. ;  and  these  are  the  intervals  to  be  used  in  reducing 
the  subsequent  observations. 

Ex.  1.  The  following  observations  of  Capella  were  made  at 
Greenwich,  January  27th,  1851 : 

A  5h.  4m.     — 


B     .     .     . 

202s. 

C     .     .    .     . 

.     .     .                 40.2 

D     .     .     .     . 

59.8 

E     .     .     .    . 

...           5      19.7 

F         ... 

.     .                      39.5 

G- 

594 

Mean  of  wires  observed,  5h.  5m.    9.8s. 
The  sum  of  the  equatorial  numbers  for  wires  B,  C,  D,  E,  F, 


THE    TRANSIT   INSTRUMENT.  57 

Gr  is  —41.443s.,  which,  divided  by  6,  gives  —6.907s.,  and,  multi- 
plied by  the  secant  of  the  declination,  45°  50'  26",  gives  —  9.91s. ; 
which,  being  applied  to  the  above  mean,  gives  5h.  4m.  59.89s. 
as  the  time  of  transit  over  the  mean  of  the  seven  wires. 

Ex.  2.  The  following  observations  of  Sirius  were  made  at 
Greenwich,  February  13th,  1851 : 

A 6h.  37m.     — 

B      ......; 

C     ......    '£H 

D W  43.7s. 

E '(&  58.2 

F M?        38      12.6 

G- 26.9 

Mean  of  wires  observed,  6h.  38m.    5.35s. 
The  sum  of  the  equatorial  numbers  for  wires  D,  E,  F,  Gr  is 

—  82.905s.,  which,  divided  by  4,  gives  -20.726s.,  and,  multi- 
plied by  the  secant  of  the  star's  declination,  16°  31'  12/x,  gives 

—  21.62s. ;  which,  applied  to  the  above  mean,  gives  for  the  time 
of  transit  over  the  mean  wire,  6h.  37m.  43.73s. 

Ex.  3.  The  following  observations  of  Spica,  Dec.  10°  22'  56", 
were  made  at  Greenwich,  February  21st,  1851 : 

A 13h.  15m.     — 

B - 

C 16        9.1s. 

D 23.1 

E      .......  37.0 

F     f-<:p';?  V-V*1.    ,  51.1 

G-    .  v;-:*H  ; •;•'•; -:-.K':I      17      5.0 

Required  the  time  of  transit  over  the  mean  wire. 

Ans.  13h.  16m.  23.01s. 

(76.)  In  the  case  of  a  star  near  the  pole,  we  must  multiply 
the  sine  of  the  equatorial  interval  by  the  secant  of  the  star's  dec- 
lination, and  we  shall  obtain  the  sine  of  the  reduction  to  the 
mean  wire  according  to  Art.  74. 

Example.  The  following  observations  of  Polaris,  at  its  upper 
culmination,  were  made  at  Greenwich,  May  30th,  1851 : 

A Oh.  38m.    — 

B     . 47 

C  56        7.0s. 


58  PRACTICAL   ASTRONOMY. 

D     .     .  ,.«*v-^fi>>     -     lh.    4m.  59.0s. 

E 13      53.0 

F 22      47.0 

G-    9^'^to-1s&to+&       31      40.0 
To  determine  the  time  of  transit  over  the  mean  of  the  seven 
wires,  the  declination  of  Polaris  being  88°  30'  38".4. 
The  reduction  for  each  wire  is  computed  as  follows : 

Wire  C.  Wire  D. 

sine  13.816s.  =  7.0020484  log.  0.002.S = 7.301 

sec.  Dec.  =  1.5851796  sec.  Dec.  =  1.585 

sine  8m.  51.70s.  =  8.5872280  log.  0.08s. =£886 

Wire  E.  Wire  F. 

sine  13.811s.  =  7.0018912  sine  27.654s.  =  7.3034239 

sec.  Dec.  =  1.5851796  sec.  Dec.  =  1.5851796 

sine  8m.  51.51s. =8.5870708     sine  17m.  45.05s.  =  8.8886035 

Wire  G. 

sine  41.438s.  =  7.4790643 
sec.  Dec.  =  1.5851796 


sine  26m.  37.92s.  =  9.0642439 

The  sum  of  these  corrections  is  — 44m.  22.86s.,  which,  divided 
by  5,  gives  —8m.  52.57s.,  which  is  the  correctio'n  to  be  applied 
to  the  mean  of  the  wires  observed  to  obtain  the  mean  of  the  sev- 
en wires.  The  mean  of  the  wires  observed  is  Ih.  13m.  53.2s. 
Hence  the  concluded  transit  over  the  mean  of  the  seven  wires 
is  Ih.  5m.  0.63s.  As  the  time  required  for  the  pole  star  to  pass 
from  one  wire  to  another  is  nearly  the  same  for  every  day  of  the 
year,  and  only  varies  in  consequence  of  a  small  change  in  the 
star's  declination,  it  is  customary,  in  regular  observatories,  to 
compute  the  intervals  for  an  assumed  value  of  the  declination, 
and  the  variation  caused  by  a  change  of  one  second  in  the  dec- 
lination. All  the  reductions  are  then  made  with  great  facility. 
(77.)  In  observing  the  sun,  the  times  of  passage  of  both  the 
first  and  second  limbs  over  the  wires  are  observed  and  set  down 
as  distinct  observations,  the  mean  of  which  gives  the  time  of 
passage  of  the  centre  across  the  meridian.  The  wires  of  the  in- 
strument are  generally  placed  by  the  maker  at  such  a  distance 
from  each  other  that  the  first  limb  of  the  sun  shall  have  passed 
all  of  them  before  the  second  limb  arrives  at  the  first  wire. 
If  only  one  limb  is  observed,  the  passage  of  the  centre  may 


THE    TRANSIT   INSTRUMENT.  59 

be  inferred  by  adding  or  subtracting  the  sidereal  time  of  semi- 
diameter  passing  the  meridian,  as  given  on  page  first  of  each 
month  in  the  Nautical  Almanac. 

Only  one  limb  of  the  moon  can  be  observed,  except  when  her 
transit  happens  to  be  within  an  hour  or  two  of  her  opposition ; 
and,  in  observing  the  larger  planets,  the  first  and  second  limbs 
may  be  observed  alternately  over  the  seven  wires-.  If  only  one 
limb  of  a  planet  is  observed,  the  ephemeris  must  be  consulted 
for  the  time  of  passage  of  its  semidiameter. 

(78.)  In  correcting  imperfect  transits  of  the  sun  and  planets, 
the  value  of  the  intervals  found,  as  for  a  star  of  the  same  decli- 
nation, must  be  increased  by  a  small  quantity.  For  if  a  fixed 
star  and  the  sun's  first  limb  were  together  at  the  first  wire,  the 
sun  would  be  behind  the  star  when  it  passed  the  second  wire, 
on  account  of  the  sun's  apparent  motion  among  the  stars.  For 
the  sun  or  a  planet,  therefore,  the  interval  found  for  a  star  must 
be  multiplied  by  the  factor 

3600+1 
3600   ' 

where  I  represents  the  hourly  increase  of 'right  ascension  in 
seconds  of  time  taken  from  the  Nautical  Almanac. 

Example.  The  following  observations  of  the  sun's  second 
limb  were  made  at  Greenwich,  February  22d,  1851 : 

A 22h.  20m.     — 

B — 

C     ."•  ."    ...     .     .  54.5s. 

D ..„..„',.     .  21        8.8 

E     .     /Y,l_.     .  22.9 

F 36.8 

G- 51.0 

Mean  of  wires  observed,  22h.  21m.  22.8s. 
The  sum  of  the  equatorial  numbers  for  wires  C,  D,  E,  F,  G 
is  —69.089s.,  which,  divided  by  5,  gives  —13.82s.,  and,  multi- 
plied by  the  secant  of  the  sun's  declination,  10°  VH/  41X/,  gives 
-  14.04s.,  which  is  the  correction  for  a  star  of  the  same  decli- 
nation.    The  sun's  hourly  increase  of  right  ascension,  Febru- 
ary 22d,  according  to  the  Nautical  Almanac,  was  9.52s.    Hence 

3600  :  3609.52  : :  14.04  : 14.08, 
which  is  the  correction  to  the  mean  of  the  wires  observed. 


60  PRACTICAL    ASTRONOMY. 

Hence   the   concluded   transit  over  the   mean   of  the   seven 
wires  is 

22h.  21m.  8.72s. 
In  order  to  facilitate  these  reductions,  it  is  convenient  to  have 

3600  +  1 
a  table  showing  the  logarithm  of  the  factor   -g^QQ     for  every 

value  of  I  from  Is.  up  to  80s. 

(79.)  The  reduction  of  an  imperfect  transit  of  the  moon's 
limb  requires  a  peculiar  method,  on  account  of  the  moon's  prox- 
imity to  the  earth. 

Let  C  represent  the  centre  of  the 
earth,  A  any  place  on  the  earth's  sur- 
face, and  M  the  centre  of  the  moon; 
then,  since  the  angular  value  of  any 
small  line  at  different  distances  is  in- 
versely as  those  distances,  the  angular 
value  of  the  moon's  hourly  motion  in 
its  orbit  from  a  place,  A,  is  to  the  an- 
gular value  of  the  same  from  C  as  CM 
to  AM.  But  CM  is  to  AM  as  the  sine 
of  CAM,  or  sine  of  ZAM,  is  to  the  sine 
of  ZCM  ;  that  is,  as  the  sine  of  the  apparent  zenith  distance  of 
the  moon  is  to  the  sine  of  the  geocentric  zenith  distance.  In 
order,  therefore,  to  reduce  an  observation  at  any  wire  to  the 
mean  of  the  wires,  the  interval  found  for  the  sun  or  a  planet 
must  be  multiplied  by  the  factor, 

sine  of  moon's  geocentric  Z .  D 
sine  of  moon's  apparent  Z .  D  ' 
or  the  entire  factor  for  the  moon  will  be 

3600  + 1     sine  of  moon's  geocentric  Z .  D 

ocrtrT"  x  ~~= f —   — r^  j.  n  -^  X  secant  of  moon's  ge- 

3600         sine  of  moon's  apparent  Z .  D 

ocentric  declination, 

where  I  is  the  hourly  increase  of  the  moon's  right  ascension  in 
seconds  of  time. 

Example.  The  following  observations  of  the  moon's  second 
limb  were  made  at  Greenwich,  February  21st,  1851 : 


THE    TRANSIT   INSTRUMENT.  61 

A 15h.,  34rn.        — 

B .  — 

C 35  9.5s. 

D 24.0 

E 38.7 

F 53.2 

a 36        8.0 

Mean  of  wires  observed,  15h.,  35m.,  38.68s. 
The  sum  of  the  equatorial  numbers  for  the  wires  observed, 
divided  by  5,  gives  —13.8178s. 

The  moon's  decimation  =14°  13'  12"  S. 

Moon's  geocentric  zenith  distance  — 65    41  50 
Moon's  apparent  zenith  distance    =66    34  10 
Moon's  hourly  increase  of  R.  A.     =  135.24s. 
The  correction  to  the  mean  of  the  wires  observed  is  then  com- 
puted as  follows : 

13.8178s.  =  1.14044 

3600  + 1  =  3735.24s.  =  3.57232 

3600  comp.  =  6.44370 

sin.  65°  41'  50"  =  9.95970 

cosec.  66    34  10   =0.03737 

sec.  14    13   12    =0.01351 

14.69s.  =  1.16704 

Subtracting  14.69s.  from  the  mean  of  the  wires  observed,  we 
obtain  the  time  of  transit  over  the  mean  of  the  seven  wires, 

15h.  35m.  23.99s. 

(80.)  We  have  hitherto  supposed  the  transit  instrument  to 
be  perfectly  adjusted — that  there  is  no  error  of  collimation — 
that  the  axis  is  perfectly  horizontal — and  that  the  middle  wire 
of  the  transit  describes  the  plane  of  the  meridian.  In  practice, 
these  adjustments  can  never  be  perfectly  made ;  but  we  make 
the  adjustments  as  complete  as  we  are  able.  We  then  com- 
pute the  amount  of  each  error,  and  apply  a  correction  to  the  ob- 
servations. 

PROBLEM. 

To  determine  the  inclination  of  the  axis  of  the  transit. 
The  spirit-level  which  rests  on  the  pivots  of  the  axis  determ- 
ines the  inclination  of  the  axis.     Above  the  glass  tube,  and  par- 


62  PRACTICAL   ASTRONOMY. 

allel  to  its  length,  is  placed  a  fine  graduated  scale,  which  indi- 
cates any  deviation  from  horizontally  by  the  air-bubble  reced- 
ing from  the  centre  toward  that  pivot  which  is  the  highest ;  but 
as  the  legs  of  the  level  may  not  be  of  exactly  equal  length,  it  is 
necessary  to  reverse  the  level  on  the  axis,  and  read  the  scale  at 
each  extremity  of  the  air-bubble  in  both  its  positions ;  that  is, 
with  the  same  end  of  the  level  on  both  the  east  and  west  pivots 
alternately.  Half  the  difference  of  the  means  of  the  two  read- 
ings will  be  the  amount  of  deviation.  It  is  customary  to  make 
several  observations  in  each  position  of  the  level,  in  order  to  di- 
minish the  effect  of  incidental  errors.  The  following  example 
will  illustrate  this  method : 

Readings  of  the  Scale. 

East  end.  West  end. 

32.3  30.0 

32.4  30.0 
32.4  30.0 

Level  reversed. 


32.6 
32.6 
32.5 

29.6 
29.5 
29.6 

194.8          sums 
32.47        means 
Difference, 

178.7 
29.78 
2.69. 

Half  the  difference  is  1.34  ;  and,  since  the  value  of  one  divis- 
ion of  the  level  is  1".25,  the  east  end  of  the  axis  is  too  high  by 
l//.67,  for  the  mean  of  the  eastern  readings  is  greater  than  the 
mean  of  the  western.  This  quantity,  divided  by  15,  will  give 
the  inclination  expressed  in  seconds  of  time. 

(81.)  Having  determined  the  inclination  of  the  axis,  the  cor- 
rection to  be  applied  to  the  time  of  observation  of  any  star  may 
be  computed  by  the  following  method : 

PROBLEM. 

To  compute  the  correction  to  the  time  of  transit  for  inclina- 
tion of  the  axis, 

Let  P  represent  the  pole  of  the  earth,  Z  the  zenith,  N  and  S  the 
north  and  south  points  of  the  horizon.  Suppose  the  transit  tel- 


THE    TRANSIT    INSTRUMENT. 


63 


escope  is  in  the  meridian,  at  the 
north  and  south  points  of  the 
horizon,  N  and  S,  but  the  axis 
is  inclined  to  the  horizon  by  a 
small  angle ;  the  telescope,  in- 
stead of  describing  the  rnerid- 
ian,  NZS,  will  describe  an  ob- 
lique circle,  NAS ;  and  the  star, 
A,  when  it  passes  through  the 
telescope,  will  be  distant  from 
the  meridian  by  the  angle  APS. 
Now,  in  the  triangle,  APS,  we 
have  sin.  PA  :  sin.  S  :  :  sin.  SA  :  sin.  P  ; 

or,  putting  b  to  represent  the  angle  S,  and  Z  the  zenith  dis- 
tance of  the  star  (b  being  supposed  to  be  a  small  angle), 

b  cos.  Z 

cos.  Dec.  :  b  :  :  cos.  Z  :  P  = ^ — , 

cos.  Dec. 

which  must  be  subtracted  from  the  observed  time  of  passage  to 
have  the  true  time,  when  the  telescope  is  inclined  to  the  west. 
"When  the  eastern  pivot  is  too  high,  the  level  error  is  considered 
negative ;  when  the  western  pivot  is  too  high,  the  level  error  is 
positive.  "? 

(82.)  The  expression  for  the  zenith  distance  of  a  star,  in 
terms  of  its  declination  and  of  the  latitude  of  the  place,  will 
vary  according  as  the  observations  are  made  to  the  south  of  the 
zenith  or  to  the  north  of  the  zenith ;  and,  in  the  latter  case, 
according  as  the  observations  are  made  above  or  below  the  pole. 
These  several  values  will  be  as  follow,  representing  the  latitude 
by  0,  and  the  declination  by  6  (see  page  139) : 

Z  —  $  —  6     -     -     if  the  observations  be  made  to  the  south ; 
Z  —  d—fi     -     -     if  to  the  north,  above  the  pole  ; 
Z  =  180°-(0  +  (S)  if  to  the  north,  below  the  pole. 

Example.  Castor  was  observed  to  pass  the  meridian  of  Green- 
wich, February  22d,  1851,  at  7h.  24m.  6.52s.,  its  declination 
being  32°  12X  32/x  N.,  and  the  error  of  level  -3x/.92;  required 
the  corrected  time  of  transit. 

The  latitude  of  Greenwich  is  51°  28'  39/x ;  therefore  Z  =  19° 
16X  7". 


64  PRACTICAL   ASTRONOMY. 

b  =  -  3". 92= 0.261s.  log.  =  9.4166 

cos.  Z  =  9.9749 

sec.  Dec.  =  0.0726 

-0.29s.  =  04641 

Therefore,  the  time  of  transit,  corrected  for  errbr  of  level,  is 
7h.  24m.  6.23s. 

PROBLEM. 

(83.)   To  determine  the  error  of  collimation. 

This  error  may  be  determined  by  a  micrometer  attached  to 
the  eye  end  of  the  telescope,  by  which,  when  the  telescope  is 
directed  toward  any  distant  object,  the  angular  distance  of  that 
object  from  the  central  wire  is  measured.  The  instrument  is 
then  reversed,  and  the  distance  of  the  same  object  from  the  cen- 
tral wire  again  measured.  Half  the  difference  of  these  meas- 
ures is  the  error  o£  collimation  for  the  middle  wire. 

(84.)  At  many  observatories  the  error  of  collimation  is  determ- 
ined, not  by  observations  of  a  distant  mark,  but  by  means  of  a 
small  transit  instrument,  mounted  at  a  short  distance  from  the 
large  transit,  and  in  the  same  meridian,  and  having  in  its  focus 
a  cross  in  the  form  of  an  acute  X.  A  reflector,  is  attached,  for 
the  purpose  of  throwing  the  light  of  the  sky  upon  the  wires,  and, 
when  the  telescopes  are  pointed  toward  each  other,  the  cross  in 
the  small  transit  is  distinctly  seen  by  looking  through  the  large 
telescope.  The  following  are  the  results  of  a  set  of  observations 
at  Greenwich :  When  the  illuminated  end  of  the  axis  was  east, 
and  the  micrometer  was  made  to  coincide  with  the  cross,  the 
reading  was  10.888r. ;  when  the  axis  was  reversed,  the  reading 
was  9.461r. ;  hence  the  reading  of  the  micrometer  for  the  true 
line  of  collimation  was  10.174r.  When  the  micrometer  was 
made  to  coincide  with  the  middle  wire,  the  reading  was  10.191r. ; 
hence  the  error  of  collimation  for  the  middle  wire  was  0.017 
revolutions  of  the  screw,  which  is  equal  to  Ox/.28  in  seconds  of 
arc.  Correcting  this  for  the  distance  of  the  middle  wire  from 
the  mean  of  the  seven  wires,  we  obtain  the  error  of  collimation 
for  the  mean  of  the  seven  wires. 

(85.)  This  error  may  also  be  determined  by  observing  the 
transit  of  Polaris,  or  any  other  close  circumpolar  star,  over  the 
first  three  wires;  and  then,  reversing  the  axis,  observing  the 


THE    TRANSIT    INSTRUMENT. 


65 


same  intervals  in  a  reversed  order.  The  wires  which  were  the 
first  three  in  the  former  position,  will  now  be  the  last  three. 
Let  each  of  the  observations  be  reduced  to  the  mean  wire,  ac- 
cording to  Art.  76 ;  then,  if  there  were  no  error  of  collimation, 
the  mean  of  the  observations  in  the  first  position  of  the  tele- 
scope ought  to  be  the  same  as  the  mean  in  the  reversed  posi- 
tion. But  if  the  two  results  differ  from  each  other,  it  must  be 
owing  to  error  of  collimation. 

(86.)  Suppose  the  telescope  does  not  move  in  the  meridian, 
NS,  but  in  a  small  circle,  AB,  par- 
allel  to  the  meridian,  and  every 
where  a  certain  number  of  seconds 
(c)  east  of  it.  Let  P  be  the  pole, 
and  C  the  place  of  the  star.  Draw 
CD  perpendicular  to  NS.  Then, 
when  the  star  passes  the  telescope, 
its  angular  distance  from  the  me- 
ridian will  be  CPD.  Now,  in  the 
triangle  CPD,  we  have 

Trig.,  Art.  211, 


Whence 


B,.  sin.  CD^sin.  PC  sin.  CPS. 
CD  c 


CPS= 


(1) 


sin.  PC     cos.  Dec. 
and  c=CPS  cos.  Dec (2) 

The  following  example  will  show  the  application  of  this  meth- 
od. At  Edinburgh  Observatory  the  transit  of  Polaris  was  ob- 
served over  two  wires ;  the  instrument  was  then  reversed,  and 
the  transit  observed  over  the  same  wires,  as  follows : 


Times  observed. 

Reduction.         | 

Times  reduced. 

Wire 
Wire 

Wire 
Wire 

h.     m. 

I.  ...  0  44 
II.    .  .  0  52 

Instrument  reversed. 

II.    .  .  1     8 
I.  ...  1  17 

33.5 
46.0 

56.0 

8.5 

7??. 

+  16 

+   8 

-  8 
-16 

23*59 
12.28 

12.28 
23.59 

k. 
I 
1 

1 
1 

in. 

0 
0 

0 
0 

57.09  j 

58.28  ; 

43.72  j 
44.91  < 

57.68s. 
44.31s. 

Column  second  shows  the  computed  reduction  to  the  mean 
wire,  according  to  Art.  76.  Column  third  shows  the  times  of 
transit  reduced  to  the  mean  wire.  The  difference  between  the 
mean  of  the  first  two  observations  and  the  last  two  is  13.37s. ; 

E 


66 


PRACTICAL   ASTRONOMY. 


half  of  which,  being  6.685s.,  represents  the  angle  GPS,  which, 
multiplied  by  the  cosine  of  the  star's  declination,  gives  the  error 
of  collimation,  0.181s.,  expressed  in  seconds  of  time,  which  is 
minus  for  the  first  position  of  the  instrument,  and  plus  for  the 
second  position. 

(87.)  There  is  another  method  of  determining  the  error  of 
collimation,  which  is  exceedingly  convenient  and  accurate.  It 
consists  in  pointing  the  telescope  vertically  downward  toward  a 
vessel  of  mercury,  and  observing  the  spider  lines  of  the  telescope 

as  reflected  from  the  surface  of  the 
mercury,  a  strong  illumination  be- 
ing thrown  upon  the  system  of  wires 
by  a  lateral  lamp.  The  rays  di- 
verging from  the  wires  at  A  issue 
in  parallel  lines  from  the  object- 
glass,  fall  upon  the  mercury,  M, 
and  are  thence  reflected  back  in 
parallel  lines  to  the  object-glass, 
which  is  enabled  to  collect  them 
again  in  its  focus.  Thus  is  formed 
a  reflected  image  of  the  system  of 
spider  lines  ;  and  if  the  axis  is  per- 
fectly horizontal,  and  there  is  no 
error  of  collimation,  the  reflected 
system  ought  to  coincide  exactly 
with  the  real  system,  as  seen  in  the  eye-piece  of  the  instrument. 
If,  however,  when  the  axis  has  been  leveled,  the  two  systems  of 
lines  do  not  coincide,  the  difference  is  twice  the  error  of  collima- 
tion, and  may  be  measured  by  the  micrometer. 

(88.)  The  preceding  observation  requires  a  peculiar  eye-piece, 

called  the  collimating 
eye  -  piece,  first  sug- 
gested by  Bohnenber- 
ger  in  1825.  The  col- 
limating eye-piece  con- 
sists of  an  ordinary 
positive  eye-piece, with 
an  aperture,  A,  cut  in 
its  side,  and  a  plane 


THE    TRANSIT   INSTRUMENT.  67 

perforated  speculum,  BB,  inserted  between  the  two  lenses,  at 
an  angle  of  45°  with  the  optical  axis  of  the  telescope,  as  repre- 
sented in  the  preceding  figure.  A  lamp  being  held  so  as  to 
throw  a  strong  light  upon  the  speculum,  the  reflected  images 
of  the  wires  may  be  seen  with  great  distinctness.  Instead  of  a 
perforated  opaque  speculum,  a  piece  of  plane  glass,  with  parallel 
faces,  without  any  perforation,  is  sometimes  used.  The  observer 
looks  through  the  plane  glass  without  difficulty,  while  sufficient 
light  is  reflected  from  the  lower  surface  to  render  the  lines  visible. 

(89.)  If  the  axis  of  the  telescope  be  not  horizontal,  half  the 
distance  between  the  middle  wire  and  its  image,  corrected  for 
error  of  level,  will  give  the  error  of  collimation  of  the  middle 
wire.  Correcting  this  for  the  distance  of  the  middle  wire  from 
the  mean  of  the  seven  wires,  we  obtain  the  error  of  collimation 
for  the  mean  of  the  seven  wires. 

(90.)  By  reversing  the  axis  of  the  transit  upon  its  supports, 
we  may  obtain  the  error  of  level,  as  well  as  of  collimation,  by 
means  of  the  micrometer.  If  the  errors  of  collimation  and  in- 
clination of  the  axis  are  both  in  the  same  direction,  the  devia- 
tion of  the  middle  wire  from  its  reflected  image  will  represent 
twice  the  sum  of  the  errors  of  collimation  and  level  ;  but  if  the 
axis  be  reversed,  the  deviation  will  be  twice  the  difference  of 
these  quantities.  Knowing  the  sum  and  difference  of  these 
quantities,  their  values  are  readily  determined. 

PROBLEM. 

(91.)  To  determine  the  correction  to  the  time  of  transit  for 
error  of  collimation. 

This  correction  is  readily  computed  by  equation  (1),  Art.  86. 
"We  have  only  to  multiply  the  error  of  collimation  by  the  secant 
of  the  declination  of  the  star. 

Ex.  The  transit  of  Castor,  Dec.  32°  12'  32"  N.,  was  observed 
at  Greenwich,  February  22d,  1851,  at  7h.  24m.  6.52s.,  the  er- 
ror of  collimation  being  —  0/x.93  ;  required  the  corrected  time 
of  transit. 

-Ox/.93^  0.062s.  log.  =  8.792 

sec.  Dec.  =  0.073 

-  0.07s.  = 


Therefore,  the  corrected  time  of  transit  is  7h.  24m.  6,45s. 


68  PRACTICAL   ASTRONOMY. 

% 

PROBLEM. 

(92.)  To  determine  the  deviation  of  the  transit  instrument 
from  the  meridian. 

First  Method. — By  the  pole  star,  or  any  close  circumpolar 
star. 

If  the  transit  telescope  revolves  on  a  horizontal  axis  in  the 
plane  of  the  meridian,  the  intervals  of  time  between  two  suc- 
cessive passages  of  the  pole  star  over  the  central 
wire  must  be  exactly  12  hours.  If  this  interval  dif- 
fers from  12  hours,  then  the  instrument  deviates 
from  the  true  meridian,  and  the  amount  of  devia- 
tion, as  measured  on  the  horizon,  may  be  computed 
as  follows : 

Let  ZPN  be  a  meridian,  and  ZAM  the  vertical 
circle  described  by  the  telescope  ;  let  ABDC  be 
the  small  circle  described  by  the  star  about  the 
pole,  P.  This  star  will  be  observed  with  the  tran- 
sit telescope  at  the  points  A  and  B  instead  of  C 
and  D. 

Let  </>=the  latitude  of  the  place  ; 

jy=the  polar  distance  of  the  star; 

a=the  angle  MZN^the  deviation  of  the  telescope  from 

the  meridian ; 
A=the  interval  between  two  successive  transits,  minus 

12  hours. 

In  the  triangle  APZ,  we  have 

sin.  PA  :  sin.  ZA  : :  sin.  AZP  :  sin.  APZ  ; 
or,  since  small  angles  are  nearly  proportional  to  their  sines, 

,         v  .-r,,,     a  cos.  ((b+p) 

sin.  p :  cos.  (4>-\-p) : :  a :  APZ  —  — 

sin.  p 

__a(cos.  </>  cos.  jo— sin.  0  sin.p) 

sin.  p 
=  a  cos.  0  cot.  p — a  sin.  <f). 

(See  Trig.,  Art.  72) 
Also,  in  the  triangle  BPZ,  we  have 

sin.  BP  :  sin.  BZ  : :  sin.  BZP :  sin.  BPZ, 
or 


THE    TRANSIT   INSTRUMENT.  69 

\  TJTYVT       &  °OS.   (0  —  2?) 

sin.  p :  cos.  (0—  p)  ::a:  BPN  = .  vy    *' 

sin.  p 

_a  (cos.  <f>  cos.  j9+sin.  0  sin.  jp) 

sin.  j» 

=  #  cos.  0  cot.  p-\-a  sin.  0. 
Therefore,  A=APC-f  BPD    =2a  cos.  0  cot.  p, 

and  #  =  « =  ~  sec-  0  c°t-  Dec.  j 

2  cos.  ^  cot.  ^?     2 

that  is,  the  azimuthal  deviation  of  the  transit  is  equal  to  half 
the  difference  between  the  observed  interval  and  12  hours,  in 
seconds,  multiplied  by  the  secant  of  the  latitude  and  the  co- 
tangent of  the  star's  declination. 

Ex.  1.  January  6th,  1850,  the  transit  of  Polaris,  Dec.  88° 
SO7  50/x,  was  observed,  sub  polo,  at  Greenwich,  at  13h.  4m. 
39.40s. ;  and  January  7th,  at  Ih.  4m.  57.62s. ;  the  observations 
being  corrected  for  collimation,  level,  rate  of  clock,  and  change 
of  right  ascension.  Required  the  azimuthal  error. 

%= 9.11s.  =  136-  65  log.  =  2.1356 
Je 

cot.  88°  30'  50"  =  8.4140 

sec.  0=0.2056 

a=  +  5".69  =  0.7552 

Since  the  time  elapsed  in  traversing  the  eastern  semicircle  is 
more  than  12  hours,  the  plane  of  the  telescope  falls  to  the  west 
of  the  true  meridian  on  the  north  horizon. 

Ex.  2.  April  23d,  1850,  the  transit  of  Polaris,  Dec.  88°  30' 
27",  was  observed  at  Greenwich  at  Ih.  3m.  34.99s. ;  and  April 
24th,  sub  polo,  at  13h.  3m.  22.55s.  Required  the  azimuthal 
error.  Ans.  +3X/.90 

The  factor  sec.  0  cot.  Dec.  is  sensibly  constant  for  Polaris  at 
each  observatory,  through  an  entire  year  or  more ;  hence  it  is 
well  to  prepare  this  factor  once  for  all.  At  Greenwich,  in  1850, 
the  rule  was  to  divide  A,  in  seconds  of  time,  by  3.206,  to  obtain 
the  azimuthal  deviation  in  seconds  of  space.  Thus,  in  Ex.  1, 
18.22s.,  divided  by  3.206,  gives  5x/.69  =  a. 

(93.)  Second  Method. — By  two  stars  differing  consider- 
ably in  declination. 

If  we  take  the  difference  between  the  observed  passages  of 
two  stars  over  the  meridian,  and  also  the  difference  of  their  com- 


70 


PRACTICAL   ASTRONOMY. 


m  V  puted  right  ascensions,  and  we  find 

these  differences  to  be  precisely 
equal,  the  instrument  will  be  ex- 
actly in  the  meridian ;  if  they  are 
not  equal,  this  inequality  shows  a 
deviation  from  the  meridian. 

Let  NZS  be  a  meridian,  VZV 
the  vertical  circle  described  by  the 
telescope,  P  the  pole,  Z  the  zenith, 
and  A  and  B  two  stars  observed  by 
S  the  transit  telescope. 

Let  0=the  latitude  of  the  place  ; 

6  and  6/  =  ih.Q  declinations  of  the  two  stars ; 

A=the  difference  of  the  observed  times,  minus  the 

difference  of  right  ascensions ; 
a=thQ  azimuthal  deviation  of  the  transit. 
In  the  triangle  ZPA,  we  have 

sin.  PA :  sin.  ZA  : :  sin.  PZA(  =  sin   AZS) :  sin.  ZPA, 

a  sin.  Z  . . 
v1) 


or      cos.  (5:  sin.  Z::  a:  ZPA = 


cos.  6 
a  sin.  ( 


—  d) 


cos.  d 
_#(sin.  0  cos.  6  —  cos.  0  sin.  6) 

cos.  (5 

=  a(sin.  0  — cos.  </>  tang.  6). 
In  the  same  manner,  we  find 

ZPB  =  #(sin.  </> — cos.  </>  tang.  d'). 
Therefore,      APB  =  «  cos.  0(tang.  6— tang.  <5'), 

APB 

cos.  0(tang.  6— tang.  6')' 
But,  by  Trig.,  Art.  76, 

-P,      sin.(A— B) 

tang.  A  — tang.  B  =  —  — ^. 

cos.  A  cos.  B 

TT  A  cos.  6  cos.  d/ 

Hence  a  = : —        — .. 

cos.  ^  sin.  (o — 6) 

The  sign  of  A  is  determined  from  the  equation 

where  T*  and  Rn  represent  the  observed  time  and  right  ascen- 
sion of  the  most  northern  star. 


THE    TRANSIT   INSTRUMENT.  71 

Example.  Aug.  18,  1850,  the  transit  of  6  Ceti  (Dec.  8°  57'  S.) 
was  observed  at  Greenwich  at  Ih.  16m.  0.95s.,  and  that  of  Po- 
laris (Dec.  88°  30'  N.)  at  Ih.  5m.  17.63s.,  the  difference  of  the 
tabular  right  ascensions  of  the  stars  being  10m.  40.39s.  Re- 
quired the  azimuthal  error. 

The  difference  between  the  observed  passages  is  10m.  43.32s.  ; 
hence  A  =  —  2.93s. 

2.93s.  =43".95  log.  =1.6430 

cos.  6  =8.4158 

cos.  d'=  9.9947 

sec.  0  =  0.2056 

cosec.  97°  27'  =  0.0037 


The  azimuthal  deviation  may,  in  like  manner,  be  found  by 
comparing  any  two  stars  differing  considerably  in  declination, 
and  whose  places  are  known  ;  but  it  is  always  best  to  employ 
Polaris,  or  some  close  circumpolar  star,  for  one  of  the  stars. 

(94.)  Third  Method.  —  By  two  circumpolar  stars  at  opposite 
culminations. 

There  are  two  stars  near  the  north  pole  which  culminate 
nearly  at  the  same  time,  one  above  and  the  other  below  the 
pole.  These  stars  are  51  Cephei  and  d  Ursse  Minoris.  The 
observation  of  these  stars  affords  one  of  the  best  methods  of  de- 
termining the  deviation  of  the  transit  from  the  meridian.  The 
method  of  reduction  is  the  same  as  in  Art.  93,  except  that  in- 
stead of  6/  —  6  we  must  put  (J'-t-d,  since  one  star  is  below  the 
pole.  The  observed  transits  must  first  be  corrected  for  the  er- 
rors of  level  and  collimation.  Then  put  A  =  the  difference  of  the 
observed  times,  minus  the  difference  of  the  right  ascensions,  neg- 
lecting the  12  hours.  The  error  in  azimuth  will  be 

A  cos.  d/  cos.  6 

ri  -  ___ 

~cos.  0sin.((J'-|-d)" 

Example.  On  the  9th  of  February,  1850,  the  transit  of  6 
Ursse  Minoris  (Dec.  86°  35'  43"),  sub  polo,  was  observed  at 
Greenwich  at  6h.  19m.  29.74s.,  and  that  of  51  Cephei  (Dec. 
87°  15X  26")  at  6h.  28m.  1.58s.  ;  the  difference  of  the  tabular 
right  ascension  of  the  stars  .being  12h.  8m.  20.99s.  Required 
the  error  in  azimuth. 


72  PRACTICAL   ASTRONOMY. 

A  =  10.85s.  =  162/x.75  log.  =2.2115 

cos.  6'  =  8.6799 

cos.  (5=8.7737 

sec.  0=  0.2056 

cosec.  6°  8'  51/x  =  0.9703 

a=+6xx.93  =  08410 

Hence  the  error  in  azimuth  is  +  6XX.93. 

Since  the  observed  interval  between  the  stars  was  too  great, 
it  is  plain  that  the  telescope  pointed  to  the  west  of  the  true  me- 
ridian on  the  north  horizon. 

PROBLEM. 

(95.)  To  compute  the  correction  to  the  time  of  transit  for 
the  error  of  azimuth. 

According  to  equation  (1),  Art.  93, 


cos.  6    ' 

that  is,  the  numerical  correction,  in  seconds  of  time,  to  each 
transit  is  equal  to  the  azimuthal  error,  expressed  in  seconds  of 
time,  multiplied  by  the  sine  of  the  zenith  distance  and  the  se- 
cant of  the  star's  declination. 

Ex.  1.  The  transit  of  Castor,  Dec.  32°  12X  32/x  N.,  was  ob- 
served at  Greenwich,  February  22d,  1851,  at  7h.  24m.  6.52s., 
the  azimuthal  error  being  —  8/x.32.  Required  the  corrected 
time  of  transit. 

The  zenith  distance  of  the  star  was  19°  16'  7/x. 

sin.  Z  =  9.5185 

sec.  6=0.0726 

8/x.32  =  0.5546s.  =  9.7440 

-0.22s.  =  9.3351 

Hence  the  time  of  transit  corrected  for  error  of  azimuth  is  7h. 
24m.  6.30s.  By  combining  the  results  of  pages  64  and  67,  we 
find  the  time  of  transit  corrected  for  errors  of  level,  collimation, 
and  azimuth  to  be 

7h.  24m.  6.52s.  -0.29s.  _  0.07s.-  0.22s.  =  7h.  24m.  5.94s. 
Ex.  2.  It  is  required  to  compute  the  corrections  for  the  fol- 
lowing observations  made  at  Greenwich,  February  22d,  1851  : 


THE    TRANSIT    INSTRUMENT. 


73 


Observed 

Error  o 

r 

Seconds 

Star. 

Declination. 

Transit, 

Collim., 
—  0".93. 

Level, 
-3".92. 

Azimuth, 
—  8".32. 

of  Transit 
Corrected. 

(3  Tauri                 

0 

28  28  N. 

h.   m.       s. 
5  15  53.56 

-0.07 

s. 

—028 

s. 
—  025 

s. 

52  96 

6  Ursae  Minoris  S.  P.  .  . 
Sirius      

86  35  N, 
16  31  S. 

6  19  16.31 
6  37  36.32 

+  1.05 
-0.06 

+3.27 
-0.10 

-6.24 
—0.54 

14.39 
35  62 

Antares  

26     5  S. 

16  19  17.27 

-0.07 

-0.06 

-0.60 

16.54 

The  errors  of  collimation,  level,  and  azimuth  being  taken,  as 
given  at  the  head  of  columns  4,  5,  and  6,  the  computed  correc- 
tions are  as  given  above,  and  the  corrected  seconds  of  transit  are 
given  in  the  last  column  above. 

(96.)  The  preceding  results  require  to  be  still  further  correct- 
ed for  the  error  of  the  clock,  and  we  shall  obtain  the  apparent 
right  ascension  of  the  object  observed.  Hence  we  have 


where 


R.A. 
T 


a— 


c= 


—  , 

COS.  O  COS.  O 

the  apparent  right  ascension  required. 

the  observed  time  of  transit,  as  shown  by  the  clock. 

\h^  correction  for  error  of  the  clock;  plus  when  the 

clock  is  too  slow. 
the  deviation  of  the  telescope  in  azimuth;  plus  when 

the  eastern  pivot  deviates  to  the  north  of  east. 
the  inclination  of  the  axis  of  the  telescope  ;  plus  when 

the  west  end  of  the  axis  is  too  high. 
the  error  in  collimation  ;  plus  when  the  mean  of  the 

wires  falls  on  the  east  side  of  the  optical  axis. 
0  =  the  latitude  of  the  place. 
d—  the  declination  of  the  star. 

(97.)  The  coefficients  of  <z,  b,  and  c,  being  of  daily  use  in  the 
reduction  of  observations,  should  be  computed  for  each  observa- 
tory. Table  IX.  furnishes  their  values  for  Washington  Observa- 
tory, by  means  of  which  the  reductions  are  made  with  great 
facility. 

Example.  It  is  required  to  compute  the  corrections  for  the 
following  observations  made  at  Washington  Observatory,  De- 
cember 30th,  1845  : 


74 


PRACTICAL    ASTRONOMY. 


Observed 

Error  of 

Seconds 

Star. 

Declination. 

Transit. 

Azimuth, 
—0.301s. 

Level, 
+0.249S. 

Collim., 
—0.085s. 

of  Transit 
Corrected. 

a  Persei  

+49   18 

ft.  m.       s. 
3   13  55.67 

+  0.083 

+  0.375 

s. 
—  0.130 

s. 

56  00 

•y  Eridani 

—  13  57 

3  51  24  14 

0  247 

+  0  155 

0  088 

23  96 

a  Tauri         .    . 

+  16   12 

4  27  39  13 

—0  121 

+  0  239 

—0088 

39  16 

a  Aurigae  

+45  50 

5     5  53.76 

+  0052 

+  0  355 

—  0  122 

54.04 

/3  Tauri  

+  28  28 

5  17     7.60 

-0.121 

+0.239 

-0.088 

7.63 

The  errors  of  azimuth,  level,  and  collimation  being  taken,  as 
given  at  the  head  of  columns  4,  5,  and  6,  the  corrections  given 
above  are  readily  found  by  employing  the  coefficients  of  table 
IX. ;  and  the  corrected  times  of  transit  are  given  in  the  last 
column. 

(98.)  Of  the  figure  and  unequal  size  of  the  pivots  of  the 
transit  instrument. 

The  pivots  of  the  horizontal  axis  of  the  transit  instrument 
ought  to  be  perfectly  cylindrical.  By  the  assistance  of  the  level 
we  can  easily  determine  whether  such  is  the  case.  For  this 
purpose  we  place  the  level  upon  the  pivots,  and  point  the  object 
end  of  the  telescope  downward,  as  low  as  possible  ;  we  then  di- 
rect the  telescope  upward,  as  far  as  the  level  will  permit.  If 
the  bubble  of  the  level  remains  stationary  during  this  rotation, 
we  may,  with  great  probability,  assume  that  the  pivots  are  cy- 
lindrical. This  conclusion,  however,  is  not  necessarily  exact ; 
for  if  the  sections  of  the  pivots  were  perfectly  equal  curves,  of 
whatever  kind,  symmetrically  placed  with  respect  to  the  axis  of 
rotation,  the  bubble  would  not  be  disturbed  during  the  revolu- 
tion of  the  telescope.  As,  however,  the  coincidence  of  all  these 
conditions  is  not  to  be  expected,  we  may  safely  assume  the 
pivots  to  be  cylindrical  when  they  will  stand  the  preceding  test. 

(99.)  If  the  pivots  are  made  perfectly  cylindrical,  but  of  un- 
equal diameters,  when  the  level  is  placed  upon  the  pivots,  and 
the  telescope  revolved,  the  bubble  will  not  change  its  position, 
but  the  inclination  of  the  axis,  shown  by  the  readings  of  the 
level,  will  be  erroneous ;  for  if  the  axis  of  rotation  were  per- 
fectly horizontal  while  the  pivots  were  unequal,  the  level  would 
indicate  an  inclination,  and  the  thickest  end  of  the  axis  would 
appear  to  be  higher  than  the  other.  Let  this  inclination  be 
accurately  measured  by  means  of  the  level.  Now  reverse  the 
axis  of  the  telescope.  If  the  largest  pivot  was  before  on  the 


f       THE    TRANSIT    INSTRUMENT.  75 

east  side  of  the  telescope,  it  will  now  be  on  the  west  side,  and 
the  inclination  of  the  axis  will  be  changed.  Let  the  inclina- 
tion be  again  accurately  measured  by  means  of  the  level.  One 
half  the  difference  between  the  level  errors  in  the  two  positions 
of  the  axis  gives  the  effect  of  the  difference  in  the  diameter  of 
the  pivots ;  and  one  fourth  the  difference  gives  the  effect  of  the 
difference  in  the  radii  of  the  pivots,  which  is  a  correction  to  be 
always  subtracted  from  the  larger  end. 

(100.)  For  example,  Mr.  Curley,  at  the  Georgetown  Observa- 
tory, in  1846,  found  that  when  the  pivot  C  of  his  transit  instru- 
ment rested  on  the  west  Y,  the  east  end  of  the  axis  appeared  to 
be  too  high  by  l/x.046  ;  but  when  the  same  pivot  rested  on  the 
east  Y,  the  east  end  of  the  axis  appeared  to  be  too  high  by  l/x.660. 
Hence  the  pivot  C  is  the  largest,  and  the  correction  to  the  level 
error  on  account  of  the  difference  in  the  diameter  of  the  pivots  is 

— -£—     —  —  0".153.     This  correction  is  to  be  applied  to  all 

level  readings  with  this  instrument,  inasmuch  as  the  level  de- 
termines only  the  inclination  of  the  tops  of  the  pivots,  while  in 
transit  observations  we  require  to  know  the  inclination  of  the 
axis  of  the  pivots. 

TRANSIT    OBSERVATIONS    RECORDED    BY    MEANS    OF    ELECTRO-MAG- 
NETISM. 

(101.)  Quite  recently  there  has  been  introduced*  a  new  meth- 
od of  recording  transit  observations  by  means  of  electro-magnet- 
ism. This  application  involves  two  contrivances  entirely  dis- 
tinct from  each  other.  The  first  is  a  method  by  wrhich  an  as- 
tronomical clock  may  be  made  to  break  the  electric  circuit  at 
the  end  of  every  second  ;  and  the  other  is  the  register,  for  re- 
cording not  only  the  beats  of  the  clock,  but  also  any  other  arbi- 
trary signals  at  the  pleasure  of  the  operator. 

1st.   The  electric  clock. 

(102.)  The  electric  circuit  may  be  broken  every  second,  by 
means  of  a  clock,  in  a  variety  of  ways.  Dr.  Locke  introduces 
into  the  astronomical  clock  a  wheel  with  60  teeth,  which  makes 
one  revolution  per  minute.  Each  tooth,  in  succession,  strikes 
against  the  handle  of  a  platinum  tilt-hammer,  AC,  weighing 
about  two  grains,  and  knocks  up  the  hammer,  which  almost 


76 


PRACTICAL    ASTRONOMY. 


immediately  falls  to  a  state  of  rest  on  a  bed  of  platinum.  The 
fulcrum,  B,  of  the  tilt-hammer  and  the  platinum  bed  rest,  sev- 
erally, on  a  small  block  of  wood.  Each  is  connected,  by  wires 
D  and  E ,  with  a  pole  of  the  galvanic  battery,  and  the  circuit  is 
alternately  broken  and  completed  by  the  rising  and  falling  of 
the  hammer.  The  circuit  is  open  about  one  tenth  of  a  second, 
and  closed  the  remaining  nine  tenths  of  each  second. 

(103.)  Professor  Bond  insulates  the  axis  of  the  escapement 
wheel,  and  also  the  axis  of  the  steel  pallets,  by  a  ring  of  shellac. 
Wires  from  the  two  poles  of  the  battery  are  connected  with 
each  axis,  so  that  when  either  pallet  comes  in  con- 
tact with  an  escapement  tooth,  the  galvanic  circuit 
is  closed ;  and  when  the  contact  is  broken  (as  it 
must  be  at  every  oscillation  of  the  pendulum),  the 
galvanic  circuit  is  opened. 

(104.)  At  the  Washington  Observatory  the  same 
object  is  accomplished  in  the  following  manner  :  A 
small  piece  of  metal,  M,  is  attached  to  the  back  of 
the  clock,  near  the  lower  extremity  of  the  pendu- 
lum, and  upon  it  is  placed  a  small  globule  of  mer- 
cury, so  that  the  index,  B,  attached  to  the  lower 
extremity  of  the  pendulum  may  pass  through  the 
globule  of  mercury  once  every  vibration.  A  wire 
from  one  pole  of  the  battery  is  connected  with  the 
supports  of  the  pendulum,  C,  and  another  wire 
from  the  other  pole  of  the  battery  connects  with 
the  metallic  support  of  the  mercury  globule.  If, 


THE    TRANSIT    INSTRUMENT.  77 

now,  the  pendulum  were  at  rest,  with  the  pointer,  B,  in  the  mer- 
cury, it  is  evident  that  the  electric  circuit  would  be  complete 
through  the  pendulum.  If,  then,  the  pendulum  be  set  in  mo- 
tion, it  will  break  the  circuit  whenever  it  passes  out  of  the  mer- 
cury, and  restore  it  again  as  soon  as  it  touches  the  mercury. 

(105.)  Mr.  Saxton  employs  a  small  tilt-hammer,  like  Dr. 
Locke,  but  he  breaks  the  circuit  by  means  of  a  small  glass  pin 
projecting  from  the  pendulum. 

ABC  represents  a  fine  platinum  wire,  mounted  upon  a  pivot 
at  B,  the  end  A  being 
somewhat  heavier  than 
the  other,  and  resting 
upon  a  metallic  bed,  D. 
At  C,  the  wire  is  bent 
so  as  to  form  an  obtuse 
angle.  The  wire  E 
goes  from  D  to  one  pole 
of  the  battery,  while 
the  wire  H,  from  the 
other  pole  of  the  battery,  communicates  with  the  metallic  sup- 
port, G-,  and  thence  with  the  wire  AB.  "When  the  end  A  of 
the  platinum  wire  rests  upon  the  support  D,  it  is  evident  that 
the  electric  circuit  is  complete.  This  apparatus  is  placed  near 
the  middle  of  the  pendulum  (a  portion  of  which,  IK,  is  repre- 
sented in  the  cut),  and  just  in  front  of  it,  so  that  the  pendulum 
may  swing  behind  it  without  obstruction.  A  small  glass  pin, 
F,  about  half  an  inch  in  length,  is  attached  to  the  pendulum 
in  such  a  position  that,  at  every  vibration  of  the  pendulum,  the 
pin  shall  slightly  impinge  upon  the  angle  C  of  the  platinum 
wire,  and  force  up  the  end  A.  As  soon  as  the  pin  has  passed 
the  point  C,  the  end  A  falls  back  again  upon  its  support,  D. 
Thus,  at  every  vibration  of  the  pendulum,  the  end  A  of  the 
platinum  wire  is  lifted  about  a  tenth  of  a  second,  and  rests 
upon  D  during  the  remaining  nine  tenths  of  the  second ;  that 
is,  the  electric  circuit  is  closed  about  nine  tenths  of  every  sec- 
ond, and  is  open  during  the  remaining  tenth. 

By  either  of  these  methods,  as  well  as  several  others,  the  elec- 
tric circuit  may  be  broken  every  second  by  means  of  a  clock. 

2d.  The  Register. 


78 


PRACTICAL    ASTRONOMY. 


(106.)  The  most  obvious  mode  of  registering  the  beats  of  the 
clock  is  upon  a  long  fillet  of  paper,  after  the  ordinary  method 
of  telegraphic  communications.  If  the  paper  be  allowed  to  run 
through  an  ordinary  Morse  registering  apparatus,  and  the  circuit 
be  broken  every  second  by  the  clock,  the  graver  will  trace  upon 
the  paper  a  series  of  lines  of  equal  length,  separated  by  short 
interruptions,  thus : 

It  is  easy  to  reverse  the  action  of  the  graver,  so  that,  when 
the  circuit  is  complete,  the  paper  shall  be  entirely  free,  and  a 
dot  be  made  by  the  breaking  of  the  circuit.  A  paper  graduated 
into  seconds  by  this  arrangement  exhibits  dots  with  long  inter- 
vening spaces,  thus : 

instead  of  long  lines  with  short  blanks,  as  shown  before. 

In  order  to  indicate  the  commencement  of  the  minute,  a  dot 
may  be  omitted  at  the  end  of  every  60  seconds.  This  is  ac- 
complished in  Dr.  Locke's  clock  by  omitting  one  tooth  in  the 
\vheel  which  breaks  the  circuit,  as  shown  at  H,  in  the  figure, 
page  76. 

(107.)  The  mode  of  using  the  register  for  marking  the  date 
of  any  event,  is  to  tap  on  a  break-circuit  key  simultaneously 
with  the  event.  The  beginning  of  the  short  line  thus  printed 
upon  the  graduated  scale  of  the  register,  fixes,  by  a  permanent 
record,  the  date  of  the  event.  Thus  A  represents  such  a  record 
printed  upon  the  graduated  paper. 

—    A 

By  tapping  upon  the  key  at  the  \nstant  a  star  is  seen  to  pass 
each  of  the  wires  of  a  transit  instrument,  the  observation  is  in- 
stantly and  permanently  recorded.  The  usual  rate  of  progress 
of  the  fillet  under  the  pen  is  about  one  inch  per  secondhand 
the  observations  are  read  off  by  means  of 
a  graduated  transparent  scale,  about  an 
inch  square,  as  represented  in  the  annexed 
cut,  consisting  of  equidistant  and  parallel 
lines,  ruled  upon  a  piece  of  glass  by  means 
of  a  diamond,  or  etched  with  fluoric  acid. 
If  the  interval  between  the  second  dots 
be  greater  than  the  breadth  of  the  scale, 


THE    TRANSIT    INSTRUMENT.  79 

the  scale  is  turned  obliquely  across  the  fillet,  until  the  first  and 
last  divisions  exactly  comprehend  the  space  between  the  two 
second  dots.  Let  the  distance  from  4s.  to  5s.,  on  the  above 
scale,  be  the  distance  on  the  fillet  between  the  fourth  and  fifth 
seconds,  and  let  the  dot,  a,  between  them  represent  the  obser- 
vation. It  appears,  by  inspection,  that  the  observation  was  re- 
corded between  4.7  and  4.8  seconds.  The  distance  of  a  from 
the  nearest  scale  division  may  be  estimated  to  tenths.  Thus 
time  is  accurately  measured  to  tenths,  and 
may  be  estimated  to  hundredths  of  a  second. 
On  some  accounts,  it  is  more  convenient  to 
employ  a  scale  consisting  of  diverging  lines, 
as  represented  in  the  annexed  cut,  so  that  the 
breadth  of  the  scale  may  always  exactly  com- 
prehend the  interval  between  the  second  dots, 
which  intervals  must  necessarily  vary  some- 
what in  length. 

(108.)  This  method  of  recording  transits  not  only  possesses  the 
advantage  of  precision,  but  also  of  performing  vastly  more  work 
in  a  given  time.  Fifteen  seconds  is  the  ordinary  equatorial  in- 
terval for  the  wires  of  a  transit  instrument ;  but  when  the  tran- 
sits are  printed  on  paper,  in  the  manner  now  described,  this  in- 
terval may  easily  be  reduced  to  two  or  three  seconds.  The  value 
of  a  night's  work  with  the  transit  instrument  is  thus  increased 
many  fold. 

To  obviate  the  inconvenience  of  a  long  fillet  of  paper,  Mr. 
Saxton  has  substituted  a  cylinder,  about  eight  inches  in  diam- 
eter and  two  feet  long,  enveloped  with  paper,  which  may  be  re- 
moved at  pleasure.  This  cylinder  is  made  to  revolve,  with  a 
uniform  motion,  upon  a  screw  axis,  so  that  the  recording  dots 
are  made  upon  a  perpetual  spiral.  One  sheet,  filled  in  this 
manner,  will  contain  about  two  hours'  work  with  a  transit  in- 
strument. 

(109.)  In  order  to  secure  the  full  advantage  of  the  preceding 
method,  it  is  important  that  the  paper  which  contains  the  reg- 
ister be  made  to  advance  with  entire  uniformity.  The  Messrs. 
Bond  have  invented  for  this  purpose  a  machine  which  they  call 
the  Spring  Governor,  consisting  of  a  train  of  clock-work  con- 
nected with  the  axis  of  a  fly-wheel.  It  has  an  escapement- 


80  PRACTICAL   ASTRONOMY. 

wheel,  into  the  teeth  of  which  pallets  are  operated  by  the  oscil- 
lations of  a  pendulum,  as  in  ordinary  clocks,  the  wheel  being  so 
connected  with  its  axis  by  a  spring  as  to  allow  the  axis  to  move 
while  the  wheel  is  detained  by  the  pallets.  The  register  is  made 
upon  a  sheet  of  paper  wrapped  round  a  cylinder. 

Professor  Airy,  in  order  to  impart  a  uniform  motion  to  the 
paper,  employs  a  large  conical  pendulum,  revolving  in  a  circle, 
whose  diameter  is  about  equal  to  the  arc  of  vibration  of  an  or- 
dinary seconds  pendulum. 

The  electric  method  of  recording  transits  has  been  employed  at 
the  Washington  Observatory  exclusively  since  December,  1849, 
and  it  is  now  used  also  at  the  Greenwich  Observatory. 

PERSONAL    EQUATION. 

(110.)  "We  frequently  find  that  two  individuals,  both  of  whom 
have  been  well  trained  in  transit  observations,  will  differ  by  a 
large  and  nearly  constant  quantity  in  observing  the  exact  mo- 
ment at  which  a  star  passes  a  transit  wire.  This  difference  is 
called  their  personal  equation;  and  an  allowance  should  al- 
ways be  made  for  it  whenever  observations,  which  have  been 
made  by  two  individuals  for  the  determination  of  absolute  time, 
are  to  be  combined.  This  equation  may  be  determined  by  either 
of  the  following  methods : 

(111.)  First  Method. — Let  one  observer  note  the  passage  of 
a  star  over  the  first  three  or  four  wires  of  the  transit  instrument, 
and  the  other  observer  note  the  same  star  over  the  remaining 
wires.  Let  each  set  of  observations  be  reduced  to  the  mean 
wire  by  employing  the  equatorial  interval  previously  determined. 
The  difference  between  the  two  mean  results  thus  obtained  is 
the  personal  equation  of  the  observers.  A  dozen  stars  observed 
in  the  course  of  an  hour,  will  furnish  this  equation  within  a  few 
hundredths  of  a  second. 

(112.)  Second  Method.' — The  same  object  may  be  accom- 
plished still  more  conveniently  by  employing  an  equatorial  tel- 
escope. Place  the  two  threads  of  the  micrometer  at  a  distance 
from  each  other  equal  to  about  ten  seconds  of  time,  and  adjust 
them  to  the  position  of  an  hour  circle.  Direct  the  telescope  upon 
a  star  near  the  meridian,  and  let  the  two  astronomers  observe 
the  passage  of  the  star  over  the  two  wires  alternately.  By 


THE    TRANSIT    INSTRUMMENT.  81 

means  of  the  tangent  screw  belonging  to  the  hour-circle,  bring 
the  star  back  again,  and  repeat  the  observation  as  many  times 
as  may  be  thought  necessary ;  suppose,  for  example,  20  times. 
At  10  of  these  observations,  the  individual  A  should  have  made 
the  observation  at  the  first  wire,  and  the  individual  B  at  the 
second ;  and  vice  versa  for  the  other  10  observations.  Let  M 
represent  the  mean  of  the  first  set  of  observations,  and  M'  the 
mean  of  the  second  set  of  observations  ;  then  will  the  personal 
equation  be 

M-M" 
2      ' 

(113.)  In  1843,  Dr.  Peterson,  at  Altona,  and  M.  0.  Struve, 
of  the  Pulkova  Observatory,  from  a  series  of  observations  made 
M  =  7.l75s.,  and  M'  =  7.581s.  Hence  their  personal  equation 
was  0.203s. 

In  the  same  manner,  the  personal  equation  between  Dr.  Pe- 
terson and  M.  Sabler,  of  the  Pulkova  Observatory,  was  found  to 
be  0.324s. 

The  personal  equation  between  the  late  Professor  Henderson, 
of  the  Edinburgh  Observatory,  and  Mr.  Wallace,  his  assistant, 
was  0.42s.  ^v; 

The  personal  equation  between  Mr.  Morton  and  Mr.  Rogerson, 
assistants  at  the  Greenwich  Observatory,  in  1851,  was  0.68s. 

The  personal  equation  between  the  Messrs.  Bond,  at  the  Cam- 
bridge Observatory,  is  0.31s. 

The  personal  equation  between  Processor  Keith,  of  the  Wash- 
ington Observatory,  and  Lieutenant  Almy,  in  1846,  was  0.36s. 

In  1823,  it  was  found  that  M.  Argelander  observed  transits  of 
stars  1.2s.  later  than  Professor  Bessel,  of  the  Konigsberg  Obser- 
vatory. 

The  same  year,  Argelander  observed  transits  0.20s.  later  than 
M.  Struve,  of  the  Dorpat  Observatory. 

Bessel  concluded,  from  numerous  comparisons,  that  in  1814 
there  was  no  personal  equation  between  himself  and  Struve ; 
that  in  1821,  Struve  observed  transits  0.8s.  later  than  himself; 
and  that  in  1823,  this  difference  amounted  to  an  entire  second. 

Bessel  also  discovered  that  when  he  employed  a  chronometer 
beating  half  seconds,  he  observed  transits  0.49s.  later  than  when 
he  employed  a  clock  beating  whole  seconds. 

F 


82  PRACTICAL   ASTRONOMY. 

This  personal  equation  is  but  another  name  for  positive  error 
in  the  estimation  of  fractions  of  a  second ;  and  it  not  only  va- 
ries with  different  individuals,  but  varies  with  the  same  indi- 
vidual at  different  times.  The  amount  of  this  error  in  skillful 
and  long-practiced  observers  is  truly  surprising.  Observations 
made  by  different  individuals  for  the  determination  of  absolute 
time  should  therefore  never  be  combined,  without  investigating 
the  personal  equation  of  the  observers. 


CHAPTER  III. 

GRADUATED  CIRCLES. 
MURAL    CIRCLE. 

(114.)  THE  mural  circle  consists  of  a  metallic  circle,  com- 
monly of  brass,  from  four  to  six  feet  in  diameter  when  intended 
for  a  large  observatory.  Its  circumference  is  graduated  into 
degrees  and  minutes,  and  these  are  subdivided  into  seconds  by 
a  vernier  or  a  reading  microscope.  It  revolves  upon  a  horizon- 
tal axis,  inserted  in  a  stone  pier,  so  situated  that  the  plane  of 
the  circle  may  coincide  with  the  meridian.  The  figure  on  the 
following  page  represents  the  mural  circle  used  for  many  years 
at  the  Greenwich  Observatory.  The  circle  aaaa  is  six  feet  in 
diameter,  of  brass,  and  connected  with  the  central  nucleus  by 
sixteen  spokes,  or  conical  radii.  A  circle  of  bracing  bars  is  in- 
terposed to  bind  the  cones  together,  half  way  between  the  outer 
ring  and  the  centre.  The  axis  is  a  cone  of  brass,  nearly  seven 
inches  in  diameter  in  front,  but  behind  only  about  half  as  much, 
and  nearly  four  feet  long. 

(115.)  The  telescope,  MM,  has  a  focal  length  of  six  feet  two 
inches,  the  aperture  is  four  inches,  and  its  common  magnifying 
power  about  150.  At  its  focus  are  five  vertical  wires,  and  a 
horizontal  stationary  one,  besides  a  micrometer  wire,  movable 
in  altitude,  whose  head  is  divided  into  100  equal  parts.  The 
telescope  is  attached  to  the  circle  at  the  centre  by  a  steel  axis, 
which  passes  through  the  proper  axis  of  motion  from  end  to 
end,  so  that  the  motion  of  the  telescope  round  its  own  axis  is 
concentric  with  that  of  the  circle.  For  the  purpose  of  fixing 
the  telescope  in  any  position  with  respect  to  the  circle,  there  are 
two  clamps,  one  at  each  end,  which  may  be  secured  to  the  ex- 
terior border  of  the  circle. 

The  limb  of  the  circle  consists  of  two  rings,  the  interior  one 
having  its  plane  parallel,  and  the  exterior  one  perpendicular  to 
the  plane  of  the  circle,  so  that,  when  united,  their  section  is 


84 


PRACTICAL    ASTRONOMY. 


represented  by  the  letter  T.     The  graduation  is  made  on  the 
broad  surface  of  the  exterior  ring.     The  divisions  are  made  upon 


a  narrow  ring  of  white  metal,  composed  of  four  parts  of  gold  to 
one  of  palladium  $  and  the  figures  which  count  the  degrees  are 
engraved  upon  a  similar  ring  of  platina.  Neither  of  these  metals 
tarnishes  in  the  least  degree.  The  degrees  are  cut  into  twelve 
parts,  or  5'  spaces,  and  are  numbered  from  the  pole  southward 
to  the  same  pole  again,  viz.,  from  0°  to  360°. 

(116.)  Placed  at  equal  distances  round  the  circle,  and  firmly 
attached  to  the  pier,  are  six  reading  microscopes,  A,  B,  C,  D, 
E,  F,  with  an  acute  cross  of  wires  at  their  foci  for  measuring 


MURAL    CIRCLE.  85 

angles  less  than  five  minutes.     Fig.  1  represents  the  appear- 


Fig.  2. 


ance  of  one  of  these  microscopes.  It  is  a  compound  micro- 
scope, consisting  of  three  lenses,  one  of  which  is  the  object  lens, 
at  L,  and  the  other  two  are  formed  into  a  positive  eye-piece, 
G-  H.  In  the  common  focus  of  the  object  lens  and  the  eye- 
piece, at  K,  is  placed  the  spider-line  micrometer,  similar  in  prin- 
ciple to  that  described  in  Art.  38.  It  consists  of  a  small  rect- 
angular frame,  across  which  are  stretched  two  spider  lines, 
forming  an  acute  cross,  and  is  moved  laterally  by  means  of  a 
screw,  whose  head  is  divided  into  60  equal  parts.  Fig.  2 
shows  the  field  of  view,  with  the  magnified  divisions  on  the  in- 
strument, as  seen  through  the  microscope.  When  the  micro- 
scope is  properly  adjusted,  the  image  of  the  divided  limb  and 
the  spider  lines  are  distinctly  visible  together ;  and,  also,  five 
revolutions  of  the  screw  must  exactly  measure  one  of  the  5' 
spaces  on  the  limb.  If  the  five  revolutions  do  not  include  the 
whole  of  one  space,  the  object  lens  must  be  screwed  up  toward 
the  image  of  the  limb,  and  the  position  of  the  microscope  ak 
tered  till  distinct  vision  is  obtained  both  of  the  spider  lines  and 
the  divisions  of  the  limb.  It  may  require  repeated  trials  before 
these  conditions  are  completely  fulfilled.  Moreover,  it  is  found 
that  changes  of  temperature  and  other  causes  produce  a  con- 
tinual variation  in  the  value  of  one  division  by  the  microscope. 
Each  of  the  microscopes  must  therefore  be  examined  from  time 
to  time,  and  allowance  made  for  error  of  runs.  It  is  usual  to 
measure  the  value  of  one  division  of  the  circle  by  each  micro- 
scope, at  several  different  parts  of  the  circle,  and  take  the  mean. 
The  following  example  is  taken  from  the  Washington  observa- 
tions of  1845 : 


PRACTICAL   ASTRONOMY. 


July  15, 1845.     Error  of  runs  of  the  six  microscopes  determ- 
ined for  four  points  of  the  circle. 


Pointer. 

A. 

B. 

c. 

D. 

E. 

F. 

Mean. 

72 
112 
240 
360 

+2.0 
2.5 
2.0 

1.8 

+  2.3 
3.1 
1.9 
2.2 

+2.5 
1.4 
1.8 
1.7 

+  0.5 
1.7 
1.0 
1.9 

+  1.2 
2.3 
2.5 
1.7 

+  0.0 
1.7 
2.0 
1.0 

+  1.417 
2.117 

1.867 
1.717 

Mean  . 

+  2.07 

+  2.37 

+  1.85 

+  1.27 

+  1.92|  +1.17  +1.779 

The  average  error  of  the  six  microscopes  for  an  arc  of  5  min- 
utes, July  15th,  was  +l/x.779.  Consequently,  smaller  arcs, 
which  are  measured  by  the  microscopes,  should  have  a  propor- 
tional part  of  this  error  applied  to  them  ;  that  is,  the  correction 
due  to  an  observation  for  error  of  runs  is 

_M.R 

5'    : 

where  R  represents  the  observed  error  of  runs,  and  M  is  the 
mean  of  the  microscopes,  omitting  the  largest  contained  multi- 
ple of  5  minutes. 

(117.)  The  axis  of  the  circle  is  made  horizontal  by  the  aid  of 
a  plumb  line,  suspended  in  front  of  the  circle,  and  viewed  by 
two  microscopes,  one  near  the  top,  and  the  other  near  the  bottom 
of  the  circle.  Or,  as  this  instrument  is  supposed  to  be  used  in 
conjunction  with  a  transit  instrument,  we  may  render  the  axis 
horizontal  by  moving  the  adjusting  screws,  so  as  to  make  a  zen- 
ith star  pass  the  middle  wire  at  the  instant  the  star  is  passing 
the  middle  wire  of  the  transit.  "We  may  also  bring  the  plane  of 
the  circle  into  the  meridian  by  selecting  a  star  near  to  the  hori- 
zon, and  moving  the  proper  screws  so  as  to  cause  it  to  pass  the 
middle  wire  at  the  same  instant  that  it  passes  the  middle  wire 
of  the  transit. 

(118.)  To  make  an  observation  with  the  mural  circle,  the  tel- 
escope is  pointed  upon  a  star  just  before  it  passes  the  meridian, 
and,  by  means  of  the  tangent  screw,  the  telescope  is  moved  in 
altitude  until  the  star  appears  bisected  by  the  horizontal  wire. 
An  index  or  pointer  shows  the  number  of  degrees,  and  the  near- 
est five  minutes,  while  the  minutes  less  than  five  and  the  seconds 
are  obtained  from  the  microscopes. 


MURAL    CIRCLE. 


87 


The  following  observations  were  made  at  Washington,  No- 
vember 28, 1845 : 


Stars. 

Pointer. 

A. 

13. 

c. 

D. 

E. 

F. 

7]  Tauri  

315  15 

0  37.5 

65.0 

55.3 

396 

75.7 

37.7 

a  Tauri  

322  40 

1  15.5 

446 

360 

186 

56.2 

147 

a  Orionis  .... 
e  Canis  Majoris  . 

331  30 

7  35 

0  22.5 
1  45.6 

52.0 

77.8 

43.0 
66.0 

26.0 
53.2 

63.3 
83.6 

21.0 
43.6 

It  is  required  to  determine  the  true  circle  readings,  the  error 
of  runs  on  an  arc  of  5X  at  the  time  of  each  of  the  preceding  ob- 
servations being  -  l/x.51 ;  -l/x.76;  — 1".81;  -l/x.83. 

The  mean  of  the  seconds  readings  by  the  six  microscopes  for 
T)  Tauri  is  51XX.80.  The  error  of  runs  for  5'  being  -lxx.51,  the 
error  for  51/x.8  will  be  —  0/x.27,  which,  subtracted  from  the  pre- 
ceding mean,  gives  52/x.07  for  the  number  of  seconds.  The 
pointer  indicates  315°  15'.  Hence  the  concluded  circle  reading 
is  315°  15'  52X'.07. 

For  a  Tauri,  the  mean  of  the  seconds  readings  by  the  six  mi- 
croscopes is  30/x.93 ;  correction  for  runs,  +  Ox/.54,  making  31x/.47. 
The  pointer  indicates  322°  40',  and  the  microscopes  give  V 
31XX.47.  Hence  the  concluded  circle  reading  is  322°  41X  31XX.47. 

For  a  Orionis,  mean  of  the  six  microscopes,  37/x.97 ;  correc- 
tion for  runs,  +  0/x.23. 

Concluded  circle  reading,  331°  30'  38X/.20. 

For  e  Canis  Majoris,  mean  of  microscopes,  61XX.63 ;  correction 
for  runs,  +  0/x.77. 

Concluded  circle  reading,  7°  37X  2/x.40. 

These  results  require  to  be  still  further  corrected  for  refrac- 
tion, which  is  furnished  by  Table  VIII. 

(119.)  To  determine  the  horizontal  point,  or  the  zenith 
point,  on  the  limb  of  the  circle. 

Point  the  telescope  upon  any  known  star  when  it  crosses  the 
meridian,  and  record  the  reading  of  the  circle.  On  the  next 
night,  observe  the  same  star  as  it  crosses  the  meridian,  by  point- 
ing the  telescope  upon  the  image  of  the  star  reflected  from  the 
surface  of  mercury.  As  the  surface  of  a  fluid  at  rest  is  horizon- 
tal, and  as  the  angle  of  reflection  is  equal  to  the  angle  of  inci- 
dence, this  image  will  be  just  as  much  depressed  below  the  hori- 
zon as  the  star  itself  is  above  it.  The  arc  intercepted  on  the 


88  PRACTICAL   ASTRONOMY. 

limb  of  the  circle,  between  the  star  and  its  reflected  image,  is 
the  double  altitude  of  the  star,  and  its  middle  point  is  the  hori- 
zontal point  of  the  circle,  allowing  for  the  difference  of  refraction 
at  the  moments  of  observation.  By  skillful  management  it  is  pos- 
sible to  observe  the  star  on  the  same  night,  both  by  reflection  and 
direct  vision,  sufficiently  near  to  the  meridian  to  give  the  horizon- 
tal point  without  risking  the  change  of  refraction  in  24  hours. 

(120.)  This  may  be  effected  in  the  following  manner :  Sev- 
eral minutes  before  the  star  in  question  comes  to  the  meridian, 
let  the  telescope  be  pointed  downward  upon  a  basin  of  mercury, 
previously  placed  in  the  proper  position  to  see  the  star  reflected 
from  its  surface.  Let  the  telescope  be  firmly  clamped,  and  the 
six  microscopes  be  read  and  registered.  When  the  star  enters 
the  field  of  the  telescope,  let  it  be  followed  by  the  micrometer 
wire  which  moves  in  altitude,  and  let  it  be  accurately  bisected 
at  the  instant  the  star  passes  the  first  vertical  wire.  Then  un- 
clamp  the  telescope  and  point  it  upward  toward  the  star  ;  and, 
by  means  of  the  tangent  screw,  let  the  telescope  be  moved  in 
altitude  until  the  star  is  brought  upon  the  fixed  horizontal  wire, 
and  let  it  be  accurately  bisected  at  the  instant  of  its  passing  the 
last  vertical  wire.  The  observer  may  then  read  the  microscopes 
at  his  leisure,  and  also  the  micrometer  of  the  telescope.  Know- 
ing the  value  of  one  revolution  of  the  screw,  the  first  observation 
is  easily  reduced  to  the  fixed  horizontal  wire,  so  that  we  have 
secured  a  reflected  observation  at  the  first  vertical  wire,  and  a 
direct  observation  at  the  last  vertical  wire.  Both  of  the  observ- 
ations are  to  be  reduced  to  the  middle  wire,  as  explained  in  Art. 
174.  The  mean  of  the  two  observations  thus  corrected  furnish- 
es the  horizontal  point  on  the  circle. 

(121.)  The  nadir  point,  and,  consequently,  the  zenith  point 
of  the  circle,  may  also  be  found  in  the  mode  described  in  Art. 
87.  When  the  telescope  is  directed  vertically  downward  upon 
a  basin  of  mercury,  and  the  reflected  image  of  the  horizontal 
wire  is  brought  to  coincide  with  its  direct  image,  the  telescope 
is  directed  toward  the  nadir,  which  is  distant  90  degrees  from 
the  horizontal  point,  or  180  degrees  from  the  zenith  point.  As 
this  observation  can  be  made  at  any  time  independently  of  the 
weather,  it  is  a  most  valuable  method,  and  in  many  observato- 
ries is  the  one  exclusively  employed. 


TRANSIT    CIRCLE. 


89 


(122.)  The  horizontal  point,  determined  by  direct  and  reflect- 
ed observations,  should  differ  exactly  90°  from  the  zenith  point, 
determined  by  the  collimating  eye-piece.  By  combining  the  two 
methods,  therefore,  we  have  the  means  of  testing  the  accuracy 
of  each  of  them.  The  following  are  the  results  of  observations 
made  at  Washington  in  1845 : 


1 

Star.                    |     Horizontal  Point. 

Zenith  Point. 

Difference. 

Aug.  1 
"      4 
«    18 
"    23 
Oct.  30 
Nov.   7 
«    18 

6  Ursae  Minoris    . 
6  Ursae  Minoris   . 
6  Ursse  Minoris   . 
a  Ursse  Minoris   . 
y  Cephei  

330     0     1.16 
330     0     1.42 
30     0     0.01 
29  57  34.17 
100     0     3.02 
120     0     8.50 
160     0  11.80 

240     0     1.30 
240     0     0.65 
300     0     1.00 
299  57  35.23 
190     0     3.22 
210     0     8.40 
250     0  11.69 

-0.14 

+  0.77 
-0.99 
-1.06 
-0.20 
+  0.10 
+  0.11 

y  Cephei  .         .  . 

v  Cephei  . 

During  the  interval  of  these  observations,  the  position  of  the 
telescope  was  repeatedly  changed,  so  that  the  horizontal  point 
was  brought  upon  different  parts  of  the  circle.  The  last  column 
in  the  above  table  shows  the  errors  of  the  observations,  combined 
with  the  error  in  the  graduation  of  the  circle  ;  yet  the  resulting 
error,  it  will  be  seen,  is  scarcely  appreciable. 

TRANSIT    CIRCLE. 

(123.)  As  the  mural  circle  has  a  short  axis,  its  position  in  the 
meridian  is  unstable,  and  therefore  it  can  not  be  relied  upon  to 
give  the  right  ascension  of  stars  with  great  accuracy.  It  was 
formerly  thought  necessary  at  Greenwich  to  have  two  instru- 
ments for  determining  a  star's  place  ;  viz.,  a  transit  instrument 
to  determine  its  right  ascension,  and  a  mural  circle  to  determine 
its  declination.  The  German  astronomers  have,  however,  com- 
bined both  instruments  in  one,  under  the  name  of  meridian  circle, 
which  is  essentially  the  transit  instrument  already  described, 
with  a  large  graduated  circle  attached  to  its  axis.  Until  re- 
cently, the  English  astronomers  have  generally  contended  that 
this  combination  was  only  suited  to  instruments  of  moderate  di- 
mensions ;  but  a  large  transit  circle  has  lately  been  constructed 
for  Greenwich  Observatory,  under  the  direction  of  Professor  Airy. 
The  telescope  has  an  aperture  of  eight  inches,  and  a  focal  length 
of  11  i  feet.  The  length  of  the  axis  between  the  extremities  of 
the  pivots  is  six  feet,  and  the  diameter  of  each  pivot  is  six  inches. 


90  PRACTICAL    ASTRONOMY. 

The  circle  is  six  feet  in  diameter,  and  is  made  of  cast  iron.  This 
instrument  has  been  in  constant  use  since  the  commencement 
of  the  year  1851,  and  the  old  transit  instrument  and  mural  circle 
have  been  abandoned. 

(124.)  The  figure  on  the  opposite  page  represents  the  transit 
circle  belonging  to  the  observatory  at  Cambridge,  Massachusetts. 
The  telescope,  T,  has  an  object-glass  of  four  and  one  eighth  inch- 
es aperture,  and  five  feet  focal  length.     The  length  of  the  axis 
between  the  shoulders  of  the  pivots  is  twenty-six  inches ;  the 
pivots  are  of  steel,  two  and  a  half  inches  in  diameter,  and  the 
same  in  length.     The  eye-piece  is  provided  with  two  microm- 
eters, one  having  a  vertical,  and  the  other  a  horizontal  move- 
ment.    Besides  th&Hisual  mode  of  illuminating  the  field  through 
the  axis,  there  are  facilities  for  illuminating  the  wires  in  a  dark 
field.     The  circles  are  four  feet  in  diameter,  being  cast  in  one 
piece,  and  are  both  graduated  on  silver,  from  0°  to  360°,  into 
five-minute  spaces.     There  are  eight  micrometer  reading  mi- 
croscopes, and  these  are  attached  immediately  to  the  granite 
piers  being  four  for  each  circle.     Four  of  these  microscopes  are 
seen  at  A,  B,  C,  and  D,  the  other  four  are  on  the  opposite  side  of 
the  piers.     These  microscopes  serve  to  bisect  diametrically  both 
circles.     The  five-minute  spaces  of  the  limbs  are  subdivided  by 
the  micrometers,  a  single  division  of  the  micrometer  head  being 
equal  to  one  second  of  arc,  and  may  be  read,  by  estimation,  to 
two  tenths  of  a  second.     The  arm,  E,  attached  to  the  pier,  sup- 
ports an  additional  microscope,  which  serves  as  a  pointer  to  in- 
dicate the  degrees  and  minutes  approximately.     There  are  fric- 
tion wheels  for  relieving  the  pressure  of  the  axis  pivots  upon  the 
Y's,  supported  by  plates  secured  to  the  piers. 

For  leveling  the  axis,  a  striding  level  is  employed,  which,  com- 
bined with  the  method  of  reflection  from  quicksilver  at  the  nadir 
point,  affords  an  independent  means  of  ascertaining  the  amount 
of  collimation  of  the  mid  wire  without  reversal  of  the  pivots. 
There  is,  however,  apparatus  for  reversing  the  instrument. 

The  object-glass  is  by  Merz,  of  Munich ;  the  mounting  by 
Simms,  of  London. 

With  this  instrument  one  observer  can,  at  the  same  time,  de- 
termine both  the  right  ascension  and  declination  of  a  star  with 


TRANSIT    CIRCLE. 


91 


92  PRACTICAL  ASTRONOMY. 

as  great  precision  as  it  can  be  done  by  two  observers  with  an 
ordinary  transit  instrument  and  a  mural  circle. 

Differences  of  decimation  recorded  by  electro-magnetism. 
(125.)  Differences  of  declination  may  be  recorded  by  means 
of  electro-magnetism.  This  is  accomplished  by  inserting  in  the 
focus  of  the  meridional  telescope  two  systems  of  spider  lines, 
one  vertical,  and  the  other  inclined  at  an  angle  of  45°.  Let 
AB  represent  the  horizontal  wire  of  the 
transit  instrument,  DE  the  middle  vert- 
ical wire,  and  FGf  a  wire  inclined  to  the 
latter,  at  an  angle  of  45°.  Let  the  tele- 
scope be  pointed  upon  a  star  as  it  ap- 
proaches the  meridian,  and  let  it  be  bi- 
sected by  the  wire  AB,  while  the  time  of 
passing  the  vertical  wire,  DE ,  is  recorded. 
Let  the  telescope  remain  firmly  fixed  in  its  position,  and  sup- 
pose a  second  star  enters  the  field  at  H,  and  traverses  the  path 
HL.  Let  the  instants  of  passing  FG  at  I,  and  DE  at  K,  be 
recorded.  Then,  if  the  angle  DCF  is  45°,  CK  (which  is  the 
difference  of  declination  of  the  two  stars)  will  be  equal  to  KI. 
The  line  KI  is  measured  by  the  time  required  for  the  star  to 
describe  this  portion  of  its  path ;  and,  in  order  to  convert  the 
observed  time  into  arc  of  a  great  circle,  we  must  multiply  it  by 
fifteen  times  the  cosine  of  the  star's  declination,  according  to 
Art.  72.  If  a  third  star  enters  the  field  at  M,  and  crosses  the 
wire  DE  at  N,  and  FGr  at  0,  then  ON  is  the  difference  of  dec- 
lination of  the  first  and  third  stars ;  and,  in  the  same  manner, 
by  observing  the  transits  of  any  number  of  stars  over  the  wires 
DE  and  FGr,  in  the  same  position  of  the  telescope,  we  shall  ob- 
tain their  differences  of  declination,  as  well  as  of  right  ascen- 
sion. In  order  to  diminish  the  errors  of  observation,  we  intro- 
duce a  large  number  of  inclined  wires,  at  intervals  of  two  or 
three  seconds  from  each  other,  as  well  as  a  large  number  of 
vertical  wires ;  and  the  times  of  transit  over  each  system  of 
wires  are  recorded  by  electro-magnetism,  as  explained  in  Arti- 
cles 101-109. 

(126.)  This  method  is  well  adapted  to  the  construction  of  a 
catalogue  of  stars,  where  it  is  proposed  to  record  the  position  of 
every  star  within  the  range  of  the  telescope.  For  this  purpose 


ALTITUDE    AND   AZIMUTH    INSTRUMENT. 


93 


the  telescope  is  firmly  clamped,  and  remains  fixed  in  its  position 
during  the  observations  of  an  entire  evening  or  night,  while  the 
observer,  sitting  with  his  eye  at  the  telescope,  has  but  to  press 
his  finger  upon  a  key  at  the  instant  a  star  is  seen  to  pass  each 
wire  of  the  two  systems  already  mentioned.  This  mode  of  ob- 
servation has  been  practiced  at  the  Washington  Observatory 
since  1849.  The  wires  for  right  ascension  are  35  in  number, 
and  are  divided  into  groups  or  fas- 
cicles of  five  each,  the  interval  be- 
tween two  wires  being  from  two 
to  three  seconds.  To  complete  a 
set  of  observations  on  any  one  fas- 
cicle requires  only  from  eight  to 
ten  seconds.  The  wires  for  differ- 
ences of  declination  are  also  35  in 
number,  and  are  arranged  in  groups 
of  five  each.  In  order  to  prevent 
any  confusion  between  observa- 
tions for  right  ascension  and  those  for  declination,  the  rule  is,  to 
observe  for  right  ascension  on  one  fascicle  of  wires  first ;  then, 
by  a  telegraphic  symbol,  to  denote  the  magnitude  of  the  star ; 
and  afterward,  to  observe  it  on  a  fascicle  of  inclined  wires  for 
declination.  The  several  fascicles  are  distinguished  from  each 
other  by  the  inequalities  of  the  intervals. 

ALTITUDE    AND    AZIMUTH    INSTRUMENT. 

(127.)  The  altitude  and  azimuth  instrument  consists  of  one 
graduated  circle  confined  to  a  horizontal  plane,  a  second  gradu- 
ated circle  perpendicular  to  the  former,  and  capable  of  being 
turned  into  any  azimuth,  and  a  telescope  firmly  fastened  to  the 
second  circle,  and  turning  with  it  in  altitude.  The  appearance 
of  this  instrument  will  be  learned  from  the  following  figure. 

EE  are  two  legs  of  the  tripod  upon  which  the  instrument 
rests ;  and,  in  close  contact  with  the  tripod,  is  placed  the  azi- 
muth circle,  FF.  One  of  the  foot  screws  has  a  contrivance  for 
giving  a  very  slow  motion  to  this  foot.  This  detached  piece 
stands  on  two  sharp  points,  besides  the  end  of  a  screw,  which, 
together,  form  an  isosceles  triangle,  having  a  gutter  in  which 
one  foot  of  the  tripod  rests ;  and  the  slowness  of  the  adjustment 


PRACTICAL   ASTRONOMY. 


ALTITUDE   AND   AZIMUTH   INSTRUMENT.         95 

depends  on  the  distance  of  this  foot  from  the  two  projecting  pins. 
This  leg  of  the  tripod  is  designed  to  be  placed  either  to  the  north 
or  south.  Above  the  azimuth  circle,  and  concentric  with  it,  is 
placed  a  strong  circular  plate,  Gr,  which  carries  the  whole  of  the 
upper  works,  and  also  a  pointer,  to  show  the  degree  and  nearest 
five  minutes  to  be  read  off  on  the  azimuth  circle ;  the  remain- 
ing minutes  and  seconds  being  obtained  by  means  of  the  two 
reading  microscopes,  C  and  D.  The  pillars  HH  support  the 
transit  axis,  I,  by  means  of  the  projecting  pieces,  LL.  The  tel- 
escope, MM,  is  connected  with  the  horizontal  axis  in  a  manner 
similar  to  that  of  the  transit  instrument.  Upon  the  axis,  as  a 
centre,  is  fixed  the  double  circle,  NN,  each  circle  being  placed 
close  against  the  telescope.  The  circles  are  fastened  together 
by  small  brass  pillars,  and  the  graduation  is  made  on  a  narrow 
ring  of  silver,  inlaid  on  one  of  the  sides,  which  is  usually  termed 
the  face  of  the  instrument.  The  reading  microscopes,  AB,  for 
the  vertical  circle  are  carried  by  two  arms,  PP,  attached  near 
the  top  of  one  of  the  pillars. 

In  the  principal  focus  of  the  telescope  are  stretched  spider 
lines,  as  in  the  transit  instrument,  and  the  illumination  is  ef- 
fected in  a  similar  manner. 

(128.)  Of  the  adjustments. 

The  horizontal  circle  is  first  to  be  leveled,  which  is  to  be  ef- 
fected in  the  same  manner  as  with  a  theodolite.  The  axis  of 
the  telescope  must  also  be  leveled,  as  in  the  transit  instrument, 
and  the  spider  lines  adjusted  for  collimation  and  verticality. 

The  meridional  point  on  the  azimuth  circle  is  its  reading 
when  the  telescope  is  pointed  north  or  south,  and  may  be  de- 
termined by  observing  a  star  at  equal  altitudes  east  and  west 
of  the  meridian,  and  finding  the  point  midway  between  the  two 
observed  azimuths ;  or  the  instrument  may  be  adjusted  to  the 
meridian,  in  the  same  manner  as  a  transit.  The  horizontal 
point  of  the  altitude  circle  is  its  reading  when  the  axis  of  the 
telescope  is  horizontal,  and  may  be  found,  as  with  the  mural 
circle,  by  alternate  observations  of  a  star  directly  and  reflected 
from  the  surface  of  mercury. 

(129.)  This  instrument  has  the  advantage  over  the  transit 
instrument  and  mural  circle,  in  its  being  able  to  determine  the 
place  of  a  star  in  any  part  of  the  visible  heavens  ;  but  we  ordi- 


96  PRACTICAL   ASTRONOMY. 

narily  require  the  place  of  a  star  to  be  given  in  right  ascension 
and  declination,  and  not  in  altitude  and  azimuth,  and  to  de- 
duce the  one  from  the  other  requires  a  laborious  computation. 
Hence  the  altitude  and  azimuth  instrument  is  but  little  used 
in  astronomical  observations,  except  for  special  purposes,  as,  for 
example,  to  investigate  the  laws  of  refraction. 

The  use  of  this  instrument  has,  however,  been  recently  re- 
vived at  the  Greenwich  Observatory.  In  the  year  1847,  an  al- 
titude and  azimuth  instrument  was  erected,  having  its  horizon- 
tal and  vertical  circles  each  three  feet  in  diameter.  The  length 
of  the  telescope  is  5  feet,  and  the  aperture  of  its  object  glass  3f 
inches. 

(130.)  The  leading  object  in  view  in  the  erection  of  this  in- 
strument was  to  obtain  observations  of  the  moon  in  portions  of 
her  orbit  where  she  could  not  be  observed  on  the  meridian.  It 
frequently  happens,  from  the  unfavorable  state  of  the  weather, 
that  the  moon  can  not  be  seen  when  she  is  on  the  meridian ; 
and  although  the  sky  may  be  perfectly  clear,  it  is  impossible  to 
see  the  moon  on  the  meridian  for  several  days  before  and  after 
her  conjunction  with  the  sun,  on  account  of  the  brightness  of  the 
solar  rays.  But  with  the  new  altitude  and  azimuth  instrument 
it  is  found  that  the  moon  may  be  observed  in  the  morning  and 
evening  when  she  is  only  an  hour  distant  from  the  sun.  In 
the  year  1851,  observations  of  the  moon  were  obtained  with  this 
instrument  on  206  days,  while  with  the  meridional  instruments 
it  was  only  observed  110  days.  Mr.  Airy  considers  these  results 
to  be  hardly,  if  at  all,  inferior  in  accuracy  to  those  obtained  by 
the  use  of  the  mural  circle. 

' 

SEXTANT. 

(131.)  The  arc  of  a  sextant,  as  its  name  implies,  contains 
sixty  degrees,  but,  on  account  of  the  double  reflection,  is  divided 
into  120  degrees.  The  figure  on  the  opposite  page  represents  a 
sextant,  the  frame  being  generally  made  of  brass  or  other  hard 
metal;  the  handle,  H,  at  its  back,  is  made  of  wood.  "When 
observing,  the  instrument  is  to  be  held  with  one  hand  by  the 
handle,  while  the  other  hand  moves  the  index,  G-.  The  arc, 
AB,  is  divided  into  120  or  more  degrees,  numbered  from  A  to- 
ward B,  and  each  degree  is  divided  into  six  equal  parts  of  10' 


THE    SEXTANT.  97 

each,  while  the  vernier  shows  10".     The  divisions  are  also  con- 


tinued a  short  distance  on  the  other  side  of  zero,  toward  A,  form- 
ing what  is  called  the  arc  of  excess.  The  microscope,  M,  is 
movable  about  a  centre,  and  may  be  adjusted  to  read  off  the  di- 
visions on  the  graduated  limb.  A  tangent  screw,  D,  is  fixed  to 
the  index,  for  the  purpose  of  making  the  contacts  more  accu- 
rately than  can  be  done  by  hand.  When  the  index  is  to  be 
moved  any  considerable  distance,  the  screw  I  must  be  loosened ; 
and  when  the  index  is  brought  nearly  to  the  required  division, 
the  screw  I  must  be  tightened,  and  the  index  be  moved  gradual- 
ly by  the  tangent  screw.  The  upper  end  of  the  index,  G-,  term- 
inates in  a  circle,  across  which  is  fixed  the  silvered  index-glass, 
C,  over  the  centre  of  motion,  and  perpendicular  to  the  plane  of 
the  instrument.  To  the  frame,  at  N,  is  attached  a  second  glass, 
called  the  horizon-glass,  the  lower  half  of  which  only  is  silvered. 
This  must  also  be  perpendicular  to  the  plane  of  the  instrument, 
and  in  such  a  position  that  its  plane  shall  be  parallel  to  the 
plane  of  the  index-glass,  C,  when  the  vernier  is  set  to  zero  on 
the  limb  AB. 

The  telescope,  T,  is  carried  by  a  ring,  K,  attached  to  a  stem, 

G- 


98  PRACTICAL   ASTRONOMY. 

which  can  be  raised  or  lowered  by  turning  a  milled  screw.  Its 
use  is  to  place  the  telescope  so  that  the  field  of  view  may  be  bi- 
sected by  the  line  on  the  horizon-glass  that  separates  the  silver- 
ed from  the  unsilvered  part.  In  the  telescope  are  placed  two 
wires,  parallel  to  each  other,  and  equidistant  from  the  centre  of 
the  telescope. 

Four  dark  glasses  of  different  depths  of  shade  and  'color  are 
placed  at  F,  between  the  index  and  horizon  glasses ;  also  three 
more  at  E,  any  one  or  more  of  which  can  be  turned  down,  to 
moderate  the  intensity  of  the  light  before  reaching  the  eye, 
when  a  bright  object,  as  the  sun,  is  observed. 

(132.)  The  principal  adjustments  of  the  sextant  are  the  fol- 
lowing : 

1.  To  make  the  index-glass  perpendicular  to  the  plane  of 
the  sextant. 

Move  the  index  forward  to  about  the  middle  of  the  limb ; 
then,  holding  the  instrument  with  the  divided  limb  from  the  ob- 
server, and  the  index-glass  to  the  eye,  look  obliquely  down  the 
glass,  so  as  to  see  the  circular  arc  by  direct  vision  and  by  reflec- 
tion in  the  glass  at  the  same  time ;  and  if  they  appear  as  one 
continued  arc  of  a  circle,  the  index-glass  is  adjusted.  If  it  re- 
quires correcting,  the  arc  will  appear  broken  where  the  reflected 
and  direct  parts  of  the  limb  meet.  As  the  glass  is,  in  the  first 
instance,  set  right  by  the  maker,  and  firmly  fixed  in  its  place, 
its  position  is  not  liable  to  alter,  except  by  violence ;  and  there- 
fore no  direct  means  are  supplied  for  its  adjustment. 

2.  To  set  the  horizon- glass  perpendicular  to  the  plane  of  the 
sextant. 

Screw  in  the  telescope,  T,  and  point  it  toward  a  star.  Move 
the  index  arm  backward  and  forward  past  the  zero  of  the  limb, 
and  if  the  two  images  of  the  star  do  not  exactly  coincide  in  pass- 
ing one  another,  turn  a  screw  at  the  top  or  bottom  of  the  hori- 
zon-glass, N,  until  this  coincidence  is  effected. 

3.  To  find  the  index  error. 

When  the  zero  on  the  index  is  set  to  zero  on  the  limb,  the 
horizon  and  index  glasses  should  be  parallel ;  and  if  the  tele- 
scope be  directed  to  a  star,  the  two  images  should  exactly  coin- 
cide. If  the  two  images  do  not  coincide,  this  deviation  consti- 
tutes what  is  called  the  index  error.  The  amount  of  the  index 


THE    SEXTANT.  99 

error  may  be  found  in  the  following  manner :  Clamp  the  index 
at  about  30  minutes  to  the  left  of  zero,  and,  looking  toward  the 
sun,  the  two  images  will  appear  either  nearly  in  contact,  or  over- 
lapping each  other.  Then  perfect  the  contact  by  moving  the 
tangent  screw,  and  call  the  minutes  and  seconds  denoted  by  the 
vernier,  the  reading  on  the  arc.  Next  place  the  index  about  the 
same  quantity  to  the  right  of  zero,  or  on  the  arc  of  excess,  and 
make  the  contact  of  the  two  images  perfect,  as  before,  and  call 
the  minutes  and  seconds  on  the  arc  of  excess,  the  reading  off  the 
arc.  Half  the  difference  of  these  numbers  is  the  index  error ; 
additive  when  the  reading  on  the  arc  of  excess  is  greater  than 
that  on  the  limb,  and  subtractive  when  the  contrary  is  the  case. 

EXAMPLE. 
Reading  on  the  arc 31X  56" 

Reading  off  the  arc    .     .     ;     .     .     31  22 

Difference 0'  34X/ 

Index  error  .     ..'  .     .....'  i     •     =  —  O7 17/7 

In  this  case,  the  reading  on  the  arc  being  greater  than  that 
on  the  arc  of  excess,  the  index  error  (17")  must  be  subtracted 
from  all  observations  taken  with  the  instrument,  untif  it  is  found, 
by  a  similar  process,  that  the  index  error  has  changed. 

4.  To  set  the  axis  of  the  telescope  parallel  to  the  plane  of  the 
sextant. 

There  are  two  parallel  wires  on  opposite  sides,  and  equidis- 
tant from  the  centre  of  the  field  of  the  telescope,  and  these  are 
usually  crossed  by  two  others.  Turn  either  pair  around  until 
they  are  parallel  to  the  plane  of  the  instrument.  Select  two 
stars  distant  from  each  other  90°  or  more,  and  bring  them  into 
contact  just  at  the  wire  of  the  telescope  which  is  nearest  the 
plane  of  the  sextant.  Fix  the  index,  and  alter  the  position  of 
the  instrument  so  as  to  make  the  objects  appear  on  the  other 
wire.  If  the  contact  still  remains  perfect,  the  axis  of  the  tele- 
scope is  in  proper  adjustment ;  if  not,  it  must  be  altered  by 
moving  the  two  screws  which  fasten,  to  the  up-and-down  piece, 
the  collar  into  which  the  telescope  screws.  This  adjustment  is 
not  very  liable  to  be  deranged. 

(133.)  To  measure  the  altitude  of  the  sun  by  reflection  from 
mercury. 


100  PRACTICAL   ASTRONOMY. 

Set  the  index  near  zero.  Hold  the  instrument  with  the  right 
hand  in  the  vertical  plane  of  the  sun,  toward  which  the  tele- 
scope should  he  pointed.  Two  images  will  be  seen  in  the  field 
of  view,  one  of  which,  viz.,  that  formed  hy  reflection,  will  ap- 
parently move  downward  when  the  index  is  pushed  forward. 
Follow  the  reflected  image  as  it  travels  downward,  until  it  ap- 
pears to  be  as  far  below  the  horizon  as  it  was  at  first  above,  and 
the  image  of  the  sun,  reflected  from  the  mercury,  also  appears 
in  the  field  of  view.  Fasten  the  index,  and,  by  means  of  the 
tangent  screw,  bring  the  upper  or  lower  limb  of  the  sun's  im- 
age, reflected  from  the  index-glass,  into  contact  with  the  oppo- 
site limb  of  the  image  reflected  from  the  artificial  horizon,  taking 
care  that  the  images  shall  be  midway  between  the  parallel  wires. 
The  angle  shown  on  the  instrument,  when  corrected  for  the  in- 
dex error,  will  be  double  the  altitude  of  the  sun's  limb  above  the 
horizontal  plane  ;  to  the  half  of  which,  if  the  semidiameter,  re- 
fraction, and  parallax  be  applied,  the  result  will  be  the  true  alti- 
tude of  the  centre. 

In  making  this  observation,  the  observer  should  move  the  in- 
strument round  to  the  right  and  left  a  little,  making  the  axis 
of  the  telescope  the  centre  of  motion.  By  this  movement,  the 
image  reflected  from  the  index-glass  may  be  made  to  sweep  the 
arc  of  a  circle,  and  will  pass  and  repass  the  image  seen  in  the 
mercury.  The  altitude  of  a  star  can  be  measured  in  the  same 
way  as  the  sun,  but  in  this  case  there  will  be  no  correction  for 
parallax  or  semidiameter  to  be  applied. 

(134.)  To  take  an  altitude  of  the  sun  by  means  of  the  nat- 
ural horizon. 

If  the  observer  is  at  sea,  the  natural  horizon  must  be  employed. 
Direct  the  sight  to  that  part  of  the  horizon  beneath  the  sun,  and 
move  the  index  till  you  bring  the  image  of  its  lower  limb  to 
touch  the  horizon  directly  underneath  it ;  but  as  this  point  can 
not  be  exactly  ascertained,  the  observer  should  move  the  instru- 
ment round  to  the  right  and  left  a  little,  making  the  axis  of  the 
telescope  the  centre  of  motion.  By  this  means  the  sun  will  ap- 
pear to  sweep  the  horizon,  and  must  be  made  to  touch  it  at  the 
lowest  point  of  the  arc. 

(135.)  To  find  the  distance  between  the  moon  and  sun,  or 
between  the  moon  and  a  star. 


THE    SEXTANT. 


101 


Hold  the  sextant  so  that  its  plane  may  pass  through  the  sun 
and  moon.  If  the  sun  be  to  the  right  hand  of  the  moon,  the 
sextant  is  to  be  held  with  its  face  upward ;  if  to  the  left  hand, 
the  face  is  to  be  held  downward.  With  the  instrument  in  this 
position,  look  directly  at  the  moon  through  the  telescope,  and 
move  the  index  forward  till  the  sun's  image  is  brought  nearly 
into  contact  with  the  moon's  nearest  limb.  Fix  the  index  by 
the  screw  under  the  sextant,  and  make  the  contact  perfect  by 
means  of  the  tangent  screw.  At  the  same  time,  move  the  sex- 
tant slowly,  making  the  axis  of  the  telescope  the  centre  of  mo- 
tion ;  by  which  means  the  objects  will  pass  each  other,  and  the 
contact  be  more  accurately  made ;  observing  that  the  point  of 
contact  of  the  limbs  must  always  be  observed  in  the  middle, 
between  the  parallel  wires.  The  index  will  then  point  out  the 
distance  of  the  nearest  limbs  of  the  sun  and  moon?  In  a  simi- 
lar manner  may  we  measure  the  distance  between  the  moon 
and  a  star. 


PRISMATIC     SEXTANT    OF     PIS- 
TOR    AND    MARTINS. 

(136.)  A  new  form  of  sex- 
tant, constructed  by  Pistor 
and  Martins,  Berlin,  Prus- 
sia, is  represented  in  the  an- 
nexed figure.  It  differs  in 
several  important  particu- 
lars from  the  common  sex- 
tant. 

1.  It  measures  any  angle 
up  to  180°.  Hence  double 
altitudes  of  objects  near  the 
zenith  can  be  taken  with 
it.  The  common  sextant  is 
limited  to  about  60°  as  the 
maximum  of  altitude.  The 
limb  of  the  instrument  is 
one  third  of  a  circle,  and 
is  graduated  from  zero,  to- 
ward the  left,  up  to  140°, 


102  PRACTICAL    ASTRONOMY. 

like  other  sextants.  For  angles  greater  than  this,  the  gradua- 
tion hegins  again  at  the  left  extremity  of  the  limb  with  110°, 
and  increases  toward  the  right  up  to  180°. 

2.  In  place  of  the  common  horizon-glass  is  substituted  a  rect- 
angular prism,  B,  the  diagonal  face  of  it  forming  a  mirror,  as 
explained  in  Art.  12. 

3.  Rays  from  the  object  seen  directly,  come  to  the  telescope 
without  passing  through  any  medium,  such  as  the  unsilvered 
part  of  an  horizon-glass.     Both  the  reflected  and  direct  images 
are  much  better  defined  than  is  usual  in  other  instruments. 

4.  The  index-mirror,  A,  is  so  attached  as  to  admit  of  ready 
reversal  for  determining  the  error  arising  from  want  of  parallel- 
ism of  its  surfaces.     Unlike  other  sextants,  it  receives  the  rays 
of  light  most  obliquely  when  the  index  is  at  zero.     In  measur- 
ing large  angles  there  is  no  confusion  or  multiplicity  of  images, 
and  objects  appear  distinct  and  well-defined  in  every  position 
of  the  index-glass. 

5.  The  colored  glasses,  C,  which  are  semicircles,  are  placed 
between  the  telescope  and  horizon-glass,  and  are  attached  to  an 
axis,  admitting  of  easy  reversal.     By  this  contrivance  the  effect 
of  any  want  of  parallelism  in  their  surfaces  is  entirely  obviated. 

6.  A  revolving  disk,  containing  several  colored  glasses  of  dif- 
ferent shades,  is  adapted  to  the  eye  end  of  the  telescope,  to  be 
used  in  taking  double  altitudes  of  the  sun. 

7.  A  diagonal  eye-prism,  E,  also  fits  the  eye-piece  of  the  tel- 
escope, so  that  the  head  of  the  observer  may  not  obstruct  the 
rays  from  an  object  in  measuring  angles  near  180°. 

8.  The  instrument  here  described  is  6  inches  radius,  and  the 
vernier  reads  to  10".     The  graduation  is  very  clear,  and  the  ar- 
rangement of  the  reading  microscope  and  ground-glass  screen 
(omitted  in  the  figure)  such  that  the  divisions  are  nearly  as 
easily  read  by  lamplight  as  by  daylight. 

The  other  parts  and  adjustments  of  the  instrument  are  simi- 
lar to  those  of  other  sextants. 

Reflecting  circles  are  also  constructed  by  Messrs.  Pistor  and 
Martins,  on  the  same  principle,  reading  with  two  verniers. 


REPEATING    CIRCLE. 


103 


REPEATING    CIRCLE. 


(137.)  The*  repeating  circle  bears  some  resemblance  to  the  al- 
titude and  azimuth  instrument  described  on  page  93,  but  it  has 
some  peculiarities  of  construction,  and  the  mode  of  using  it  is 
peculiar.  The  following  figure  represents  a  repeating  circle,  as 


E 


employed  by  Borda.  The  instrument  rests  upon  a  strong  tripod, 
with  feet  screws,  AAA ;  and  a  steel  spindle,  about  15  inches 
long,  has  one  end  inserted  into  the  middle  of  the  tripod,  to  which 
it  is  perpendicular.  A  hollow  brass  pillar,  D,  turns  freely  round 
this  spindle,  and  sustains  the  weight  of  the  upper  circle,  with 
its  telescopes. 


104  PRACTICAL   ASTRONOMY. 

The  azimuth  circle,  BB,  is  connected  with  the  pillar,  and  re- 
volves with  it,  while  the  divisions  are  read  by  a  vernier  attached 
to  the  tripod.  A  screw  apparatus  at  C,  attached  to  one  of  the 
branches  of  the  tripod,  clamps  the  azimuth  circle,  or  allows  it  a 
quick  or  slow  motion  at  pleasure. 

To  the  top  of  the  pillar,  D,  is  fixed  a  horizontal  brass  bar, 
with  two  supporters  at  right  angles  to  it,  in  the  tops  of  which  are 
centered  the  ends  of  a  horizontal  axis,  round  which  the  whole  of 
the  upper  part  of  the  instrument  may  be  turned,  so  as  to  bring 
the  plane  of  the  circle  into  any  position  which  may  be  required. 
The  centre  work  of  the  upper  circle,  EE,  is  made  fast  to  the 
middle  of  the  horizontal  axis,  which  it  crosses  at  right  angles  ; 
and  at  its  remote  end  is  placed  a  counterpoise,  which  balances 
the  circle  and  telescopes.  The  circle,  EE,  has  an  index  with 
four  branches,  whose  verniers  subdivide  the  circle  to  10X/. 

The  instrument  has  two  telescopes,  GrGr',  HHX,  one  in  front 
of  the  circle  and  the  other  behind  it ;  and  parallel  to  the  latter 
is  placed  the  level,  K.  The  front  telescope  moves  freely  on  a 
spindle,  within  the  axis  of  the  circle.  The  back  telescope  is  a 
little  below  the  axis  of  the  circle,  while  the  level  is  a  little  above 
it,  and  both  revolve  on  a  collar  which  works  on  the  outside  of 
that  axis.  These  can  be  fixed  in  any  position  by  a  clamp,  which 
embraces  the  back  edge  of  the  circle.  The  circle  turns  freely 
about  its  axis,  carrying  telescopes,  level,  etc.,  without  altering 
their  position  in  respect  to  itself.  There  is  a  clamp  for  fixing 
the  circle,  and  a  tangent  screw  for  slow  motion. 

(138.)  By  means  of  the  two  motions  round  a  vertical  and  hori- 
P  zontal  axis,  the  plane  of  the  circle 

/  may  be  made  to  pass  through 

any  two  points  whose  angular 
,..--s   distance  is  required  to  be  meas- 
ured.     Let  P  and  S  represent 
two  objects  whose  distance  from 
each  other  is  to  be  measured,  and 
let  GrNHX  represent  the  circle  ad- 
justed, so  that  its  plane  passes 
r* ;  through  them.      Fix  the  front 

telescope,  HHX,  at  the  zero  of  the  graduation  in  H,  and  turn  the 
circle  about  its  axis  until  the  telescope  HHX  is  directed  exactly 


REPEATING    CIRCLE.  105 

upon  the  object  S.     Clamp  the  circle  in  this  position,  and  point 
the  back  telescope,  GrGr',  upon  the  object  P.     The  angle  PCS 
will  be  measured  by  the  arc  GH,  intercepted  between  the  lines 
CP  and  CS.     Unclamp  the  circle,  and  turn  it  until  the  back 
telescope,  GrGr/,  is  pointed  toward  S.     The  front  telescope  will 
now  come  into  the  position  CL ;  the  zero  of  the  graduation, 
which  was  before  at  H,  will  be  removed  to  L.     Again  clamp  the 
circle,  release  the  front  telescope,  and  direct  it  toward  the  object 
P.     The  arc  GrHL  will  be  twice  the  arc  required  to  be  measured. 
Repeat  this  double  observation,  starting  again  from  the  point  Gr  ; 
that  is,  turn  the  circle  with  its  two  telescopes  until  the  front  tel- 
escope is  pointed  upon  the  object  S.     The  zero  of  the  graduation 
will  now  be  found  at  M.     Detach  the  back  telescope,  and  point 
it  again  upon  the  object  P ;  the  arc  GrM  will  be  three  times  the 
arc  GrH.     Unclamp  the  circle,  and  turn  it  until  the  back  tele- 
scope is  pointed  upon  S  ;  the  zero  of  the  graduation  will  now  be 
found  at  N.     Again  clamp  the  circle,  release  the  front  telescope, 
and  direct  it  toward  the  object  P.     The  arc  GrN,  which  may  be 
read  upon  the  limb,  will  be  four  times  the  arc  required.     By  re- 
peating the  observation  ten  times,  we  shall  obtain  ten  times  the 
angle  sought.     It  is  not  necessary  to  read  the  graduation  after 
each  observation ;  it  is  sufficient  to  read  the  resulting  arc  after 
the  observations  are  concluded,  and  divide  the  final  arc  by  the 
number  of  observations. 

(139.)  Suppose  these  ten  observations  should  bring  the  front 
telescope  back  to  the  zero  of  the  graduation  from  which  we 
started,  then  each  arc  would  be  equal  to  36°  ;  and  this  result 
would  not  be  affected  by  any  error  in  the  graduation  of  the  cir- 
cle. It  is  not  to  be  expected  that  the  telescope  will,  in  practice, 
be  brought  round  exactly  to  the  zero ;  but  it  should  be  brought 
round  as  near  to  zero  as  can  be  done  by  the  continued  repeti- 
tion of  the  angle  PCS  ;  then,  dividing  the  result  by  the  number 
ot  repetitions,  the  effect  of  any  error  in  the  graduation  of  the 
circle  will  be  greatly  diminished,  if  not  entirely  destroyed. 

In  a  similar  manner  may  the  zenith  distance  of  any  celestial 
body  be  measured,  by  employing  the  spirit-level  attached  to  the 
back  telescope  to  indicate  a  horizontal  line. 

(140.)  The  chief  advantage  contemplated  in  the  invention  of 
the  repeating  circle  was  the  annihilation  of  errors  of  graduation ; 


106  PRACTICAL   ASTRONOMY. 

but  the  great  improvements  which  have  been  made  in  recent 
years  in  graduating  circles  have  rendered  this  an  object  of  minor 
importance,  while  this  instrument  is  liable  to  some  serious  errors 
of  its  own,  so  that  the  repeating  circle  is  at  present  much  less 
used  than  formerly. 


CHAPTER  IY. 


THE  DIURNAL  MOTION. 

(141.)  IF,  upon  a  clear  evening,  we  carefully  watch  the  ap- 
pearance of  the  heavenly  bodies  for  a  sufficient  period,  we  shall 
find  that  they  slowly  change  their  places  with  respect  to  the  ho- 
rizon. Each  star  appears  to  describe,  as  far  as  its  course  lies 
above  the  horizon,  a  circle  in  the  sky ;  but  these  circles  are  not 
all  of  the  same  magnitude.  The  apparent  relative  situations  of 
the  stars  among  each  other  remain  unchanged ;  but  all  the 
stars  seem  to  revolve  with  a  uniform  motion  from  east  to  west, 
as  if  they  were  attached  to  the  internal  surface  of  a  vast  hollow 
sphere,  having  the  spectator  in  its  centre,  and  turning  around 
an  axis  inclined  to  the  horizon,  so  as  to  pass  through  a  fixed 
point  called  the  pole.  This  apparent  rotation  of  the  heavens  is 
called  the  diurnal  motion. 

(142.)  Let  C  be  the  place  of  the  spectator,  Z  his  zenith,  and 
N  his  nadir.  Let  PCP'  be  the 
axis  about  which  the  diurnal 
motion  is  apparently  performed, 
P  the  elevated  pole,  and  Px  the 
depressed  pole  of  the  heavens. 
Then  HMO,  a  great  circle  of  the 
sphere,  whose  poles  are  Z  and  N, 
will  be  his  celestial  horizon,  PO 
will  be  the  altitude  of  the  pole, 
OPZEH  will  be  his  meridian; 
and  ELQ,,  a  great  circle  perpen- 
dicular to  PPX,  will  be  the  celestial  equator.  Also,  if  S  repre- 
sents the  position  of  any  star,  and  PSPX  be  a  great  circle  passing 
through  it,  then  LS  will  be  the  declination,  and  PS  the  polar 
distance  of  the  star,  and  BSD  will  be  the  diurnal  circle  it  will 
appear  to  describe  about  the  pole.  0  and  H  are  the  north  and 
south  points,  e  and  w  are  the  east  and  west  points  of  the  hori- 


108  PRACTICAL   ASTRONOMY. 

zon.  Also,  if  we  draw  the  vertical  circle  ZSMN,  OM  will  be 
the  azimuth  of  the  star,  reckoned  from  the  north  point,  MS  its 
altitude,  and  ZS  its  zenith  distance. 

The  angle  ZPS,  which  the  circle  PSP'  makes  with  the  me- 
ridian PZPX,  is  called  the  Jwur  angle  of  the  star  S. 

(143.)  The  circles  thus  drawn  form  a  number  of  spherical 
triangles,  the  relations  of  whose  sides  and  angles  may  be  de- 
termined by  spherical  trigonometry.  When  the  place  of  only 
one  celestial  object  on  the  sphere  is  concerned,  it  may  be  de- 
termined from  the  triangle  PZS. 

In  the  triangle  PZS,  Z  represents  the  zenith,  P  the  elevated 
pole,  and  S  the  star,  sun,  or  other  celestial  object.  In  this  tri- 
angle the  sides  are,  1st.  PZ,  which,  being  the  complement  of 
PO,  the  altitude  of  the  pole,  is  the  complement  of  the  latitude 
of  the  place,  and  is  called  the  co-latitude ;  2d.  PS,  the  polar 
distance,  or  the  complement  of  the  declination  of  the  star ;  and, 
3d.  ZS,  the  zenith  distance,  which  is  the  complement  of  the  al- 
titude of  the  star.  If  the  object  be  situated  on  the  side  of  the 
equator  opposite  to  that  of  the  elevated  pole,  PS  will  be  greater 
than  90°. 

In  the  same  triangle  the  angles  are,  1st.  ZPS,  the  hour  angle 
of  the  star  from  the  meridian ;  2d.  PZS,  which  is  the  azimuth 
of  the  star  measured  from  the  north  point,  and  is  the  supplement 
of  HZS,  the  azimuth  measured  from  the  south  point ;  and,  3d. 
The  angle  PSZ,  which  is  called  the  parallactic  angle. 

The  sides  and  angles  of  this  triangle,  therefore,  represent  the 
following  six  astronomical  magnitudes :  1st.  The  co-latitude  of 
the  place  of  observation ;  2d.  The  polar  distance  of  the  star ; 
3d.  Its  zenith  distance ;  4th.  Its  hour  angle ;  5th.  Its  azimuth 
from  the  north  point ;  and,  6th.  Its  parallactic  angle ;  and  when 
any  three  of  these  magnitudes  are  given,  the  others  may  be 
computed. 

_^,^.  gj  PROBLEM. 

(144.)  To  find  the  altitude,  azimuth,  and  parallactic  angle 
of  a  star,  its  polar  distance  and  hour  angle  being  given,  as 
well  as  the  latitude  of  the  place. 

Let  P  be  the  pole,  Z  the  zenith,  S  the  place  of  the  star,  and 
HO  the  horizon. 


THE    DIURNAL    MOTION.  109 

Then  z 

P0=the   latitude,  which   we 

will  represent  by  0  ; 
PZ  =  the  co-latitude  =  90°  -  0  ; 
PS  =  the  polar  distance  of  the 
star; 


—  6,  where  6  represents  the  star's  decimation  ; 
ZS=  zenith  distance  of  the  star,  which  we  represent  by  Z  ; 
ZPS=the  star's  hour  angle,  which  we  represent  by  P  ; 
PZS=the  azimuth  of  the  star,  counted  from  north,  which  we 

represent  by  A. 

In  the  spherical  triangle,  PZS,  are  given  two  sides,  PS  and 
PZ,  with  the  included  angle,  to  find  the  other  parts. 

Let  fall  the  perpendicular  SM  upon  PZH  ;  then,  by  Napier's 
rule, 

R.  cos.  P  =  tang.  PM  cot.  PS. 
Therefore,        tang.  PM=cos.  P  tang.  PS 

=  cos.  P  cot.  6    ........   (1) 

But  ZM  =  PM-PZ 


Then,  by  Trig.,  Art.  216, 

sin.  PM  :  sin.  ZM  ::  tang.  SZM  :  tang.  SPM  ; 
that  is,    sin.  PM  :  cos.(PM  +  0)  ::  tang.  A  :  tang.  P  ; 

::cot.  Prcot.  A  .....  (2) 
Also,  Trig.,  Art.  216, 

cos.  PM  :  cos.  ZM  ::  cos.  SP  :  cos.  SZ  ; 
that  is,     cos.  PM  :  sin.(PM  +  </>)  ::  sin.  6  :  cos.  Z  .....  (3) 

Also,     sin.  ZS  :  sin.  P  ::  sin.  PZ  :  sin.  PSZ  ; 
that  is,     sin.  Z  :  sin.  P  ::cos.  0  :  sin.  parallactic  angle   .  (4) 

When  the  star  has  south  declination,  cot.  6  in  Eq.  1  will  be 
negative,  and  PM  must  be  taken  in  the  second  quadrant. 

Ex.  1.  Find  the  altitude,  azimuth,  and  parallactic  angle  of 
Aldebaran  (Dec.  16°  13'  N.),  to  an  observer  at  New  York, 
latitude  40°  42'  N.,  when  the  star  is  three  hours  east  of  the 
meridian. 

By  equation  (1), 

cos.  45°  =9.849485 

cot.  16°  13/  =0.536342 

PM  =     67°  38'  31"  tang.  =  0.385827 


110  PRACTICAL   ASTRONOMY. 

By  equation  (2), 

<j>=     40°  42' 

PM+</>=  108°  20'  31"      cos.  =  9.497879 

cot.  P  =  0.000000 

cosec.  PM  =  0.033941 

Azimuth=S.  71°  12'  30"  E.  cot.  =  9.531820 

By  equation  (3), 

sin.  (PM  +  P)  =  9.977356 

sin.  6=9.446025 

sec.  PM  =  0.419767 

Zenith  distance  =     45°  49'  27"       cos.  =  9.843148 
or       Altitude  =     44°  10'  33" 

By  equation  (4), 

sin.  45°  =  9.849485 

cos.  0=9.879746 

cosec.  Z  =  0.144357 

Parallactic  angle  =     48°  22'  sin.  =  9.873588 

Ex.  2.  Find  the  altitude  and  azimuth  of  Regulus  (Dec.  12° 
42'  N.),  to  an  observer  at  Washington,  latitude  38°  53'  N.,  when 
the  hour  angle  of  the  star  is  3h.  15m.  20s.  W. 

Ans.  Its  altitude  =      39°  38'    0", 

azimuth =S.  72°  28'  14"  W. 

Ex.  3.  Find  the  altitude  and  azimuth  of  Fomalhaut  (Dec. 
30°  25X  S.),  to  an  observer  at  Cambridge,  latitude  42°  22X  N., 
when  the  hour  angle  of  the  star  is  2h.  5m.  36s.  E. 

Ans.  Its  altitude  =      11°  41X  37", 

azimuth  =  S.  27°  18X  407/  E. 

Ex.  4.  Find  the  altitude  and  azimuth  of  a  Ursae  Majoris 
(Dec.  62°  33'  N.),  to  an  observer  at  Philadelphia,  latitude  39° 
57'  N.,  when  the  hour  angle  of  the  star  is  5h.  17m.  40s.  E. 

Ans.  Its  altitude  =       39°  24', 
z  azimuth =N.  35°  54'  E. 

(145.)  When  only  the  parallactic 
angle  is  required,  it  may  be  com- 
puted without  finding  the  altitude 
or  azimuth,  as  follows : 
ir~  ~~°        Draw  ZN  perpendicular  to  PS, 

and  represent  PN  by  x.     Then,  by  Napier's  rule, 


THE    DIURNAL    MOTION.  Ill 

R.  cos.  P  =  tang.  PN  cot.  PZ, 
or  cos.  P  =  tang.  x  tang.  0  ; 

that  is,  tang,  x  =  cos.  P  cot.  0  .........  (1) 

Again,  by  Spher.  Trig.,  Art.  216, 

sin.  NS  :  sin.  PN  ::  tang.  P  :  tang.  S, 

sin.  PN  tang.  P     sin.  x  tang.  P         /ox 
or     tang.  par.  ang.=  -  :  —  ——  -  —  =  —   —  -  --  ^r—  .  .  (2) 

sin.  NS  cos.(x+6) 

Example.  Required  the  parallactic  angle  for  Washington  Ob- 
servatory, latitude  38°  53'  33",  the  moon's  hour  angle  being  50° 
and  Declination  21°  N. 

By  formula  (1), 

cot.  0  =  0.093297 
cos.  P  =  9.808067 


x  =  38°  32'  55"  tang.  =  901364 

By  formula  (2), 

sin.  x  =9.794612 

tang.  P  =  0.076186 

sec.  59°  32'  55"          =  0.295157 

Par.  angle=55°  41'  24"  tang.  =0.165955 

As  the  parallactic  angle  is  frequently  required  in  many  com- 
putations, Table  XVII.  has  been  constructed  for  "Washington  Ob- 
servatory by  the  preceding  method,  except  that  instead  of  the 
geographical  latitude,  the  geocentric  latitude,  38°  42X  25",  has 
been  used.  See  Art.  208. 

(146.)  Corollary.  By  the  same  method  we 
may  compute  the  distance  between  two  stars 
whose  right  ascensions  and  declinations  are 
known. 

Let  P  be  the  pole,  and  S  and  Sx  two  stars 
whose  places  are  known.  Then  PS  and  PS'  will 
represent  their  polar  distances,  and  SPS'  will  be 
the  difference  of  their  right  ascensions.  Draw 
SM  perpendicular  to  PS'  produced.  Then 

R.  cos.  P=tang.  PM  cot.  PS. 

Therefore,        tang.  PM  =  cos.  P  tang.  PS. 

Also,  S'M^PM-PS'. 

And         cos.  PM  :  cos.  S'M  :  :  cos.  PS  :  cos.  S'S. 

Ex.  1.  Required  the  distance  from  Aldebaran,  R.  A.  4h.  27m. 


112  PRACTICAL   ASTRONOMY. 

25.94s.,  Polar  distance  73°  47X  33XX.3,  to  Sirius,  R.A.  6h.  38m. 
37.62s.,  Polar  distance  106°  31X  lxx.8. 
P=2h.  llm.  11.68s.  =  32°  47X  55XX.2         cos.  =  9.9245786 
PS=106°  31X    lxx.8       tang.  =  0.5279175 
PM  =  109°  25X  54XX.55     tang.  =  0.4524961 
PSX=  73°  47X  33XX.3 


SXM  =  35°  38X  21XX.25  cos.  =  9.9099313 

PS =106°  31X    lxx.8  cos.  =  9.4537809 

PM=109°  25X  54XX.55  sec.  =  0.4779669 

SSX  =  46°    0X42XX.3  cos.  =  9.8416791 

Ex.  2.  Required  the  distance  from  Regulus,  R.  A.  lOh.  Om. 
29.11s.,  Polar  distance  77°  18X  41XX.4,  to  Antares,  R.  A.  16h. 
20m.  20.35s.,  Polar  distance  116°  5X  55XX.5. 

Ans.  99°  55X  44XX.9. 

PROBLEM. 

(147.)  To  find  the  altitude  and  azimuth  of  a  star  when  it 
is  six  hours  from  the  meridian. 

If  the  star  S  be  six  hours  from  the 
meridian,  then  the  angle  ZPS  =  90°  ; 
the  hour  circle,  PE,  intersects  the 
horizon  in  the  east  point,  E  ;  and 
the  angle  PEG  is  equal  to  the  lati- 
tude  of  the  place.  Draw  the  verti- 
cal circle  ZSM.  Then,  in  the  right-angled  spherical  triangle 
ESM,  by  Napier's  rule, 

R.  sin.  SM  =  sin.  E  sin.  ES; 

that  is,  sin.  altitude  =  sw.  <f>  sin.  6 (1) 

Also,                   R.  cos.  E  =tang.  EM  cot.  ES  ; 
that  is,                 tang.  EM.=tang.  ES  cos.  E, 
or  cotang.  azimuth  — t^n^.  6  cos.  0 (2) 

Ex.  1.  In  Lat.  41°  18X  N.,  when  the  sun  has  18°  25X  N.  decli- 
nation, what  is  his  altitude  and  azimuth  at  six  o'clock  in  the 
morning  ? 

By  formula  (1), 

sin.  41°  18'  =9.819545 

sin.  18°25X  =9.499584 

Altitude  =     12°    2X  6XX      sin.  =  9.319129 


THE    DIURNAL    MOTION.  113 

By  formula  (2), 

tang.  18025X  =9.522417 

cos.  41°  18'  =9.875793 

Azimuth =N.  75°  57' 19"  E.  cot.  =  9.398210 

Ex.  2.  Find  the  altitude  and  azimuth  of  Regulus  (Dec.  12° 
42'  N.)  to  an  observer  at  Philadelphia,  Lat.  39°  57<  N.,  when  the 
star  is  six  hours  past  the  meridian. 

Ans.  Its  altitude  =         8°    6'  56", 

azimuth=N.  80°  11'  54"  W. 

Ex.  3.  Find  the  altitude  and  azimuth  of  Capella  (Dec.  45° 
50'  N.)  to  an  observer  at  Cambridge,  Lat.  42°  22'  N.,  six  hours 
before  the  star  comes  to  the  meridian. 

Ans.  Its  altitude  =     28°  54'  23", 
azimuth=N.  52°  44/28//  E. 

PROBLEM. 

(148.)  To  find  the  altitude  and 
hour  angle  of  a  star  ivhen  it  is  upon 
the  prime  vertical. 

Let  ZSE  be  the  prime  vertical, 
and  S  the  position  of  the  star.  Draw 
the  hour  circle  PS.  The  angle  PZS 
will  be  a  right  angle,  and  we  shall  have,  by  Napier's  rule, 

R.  cos.  P  =  tang.  PZ  cot.  SP ; 

that  is,  cos.  P  =  cot.  </>  tang  6 (1) 

Also,  R.  cos.  SP=cos.  SZ  cos.  PZ, 

cos.  SP 


or  cos.  SZ  = 


cos.  PZ' 


that  is,  sin.  altitude  =  .         (2) 

sin.  0 

Ex.  1.  Find  the  altitude  and  hour  angle  of  Aldebaran  (Dec. 
16°  13X  N.)  when  it  is  exactly  east  of  an  observer  at  New  York, 
Lat.  40°  42'  N. 

By  formula  (2), 

sin.  16°  13'  =9.446025 

sin.  40°42X  =9.814313 

Altitude  =  25°  21X  27"  sin.  =  9.631712 
H 


114  PRACTICAL   ASTRONOMY. 

By  formula  (1), 

tang.  16°  13'  =9.463658 

cot.  40°  42'  =0.065433 

70°  14'  12"  cos.  =  9.529091 
=  4h.  40m.  56.8s. = hour  angle. 

Ex.  2.  Find  the  altitude  and  hour  angle  of  Yega  (Dec.  38° 
38'  N.),  when  it  is  exactly  west  of  an  observer  at  Cambridge, 
Lat.  42°  22'  N. 

Aw.  Altitude     =  67°  53'  37", 
Hour  angle =lh.  55m.  12s. 

PROBLEM. 

(149.)  To  find  the  amplitude  and  hour  angle  of  a  star  when 
it  is  in  the  horizon. 

Let  PEP'  represent  the  hour  circle 
which  is  six  hours  from  the  meridian, 
and  which  intersects  the  horizon  in 
the  east  point,  E.      Let  S  or  S'  be 
|0   the  position  of  a  star  in  the  horizon, 
and  through  S  draw  the  hour  circle 
PSP' ;  also,  through  S'  draw  the  hour 
circle  PS'P'.     Then,  in  the  right-an- 
gled spherical  triangle  EMS  or  EM'S', 
EM  or  EM'=the  distance  of  the  star  from  the  six  o'clock 

hour  circle ; 

MS  or  M'S'=:the  star's  declination; 
ES  or  ES'=the  star's  amplitude; 

=the  complement  of  the  star's  azimuth ; 
MES  =  M'ES'  =  the  complement  of  the  latitude. 
Now,  by  Napier's  rule, 

R.  sin.  MS  =  sin.  ES  sin.  MES, 
or  sin.  ES=sin.  MS  cosec.  MES  ; 

that  is,     sin.  amplitude  =  cos.  azimuth  —  sin.  6  sec.  </>  .  .  (1) 
Also,          R.  sin.  EM = tang.  MS.  cot.  MES, 

or  sin.  EM = tang.  6  tang.  0 (2) 

P  =  6  hours  +EM, 

where  P  represents  the  time  from  the  star's  rising  to  its  passing 
the  meridian. 

Ex.  1.  Find  the  amplitude  and  hour  angle  of  Arcturus  (Dec. 


THE    DIURNAL    MOTION.  115 

19°  57'  N.)  when  it  rises  to  an  observer  at  New  York,  Lat.  40° 
42' N. 

By  formula  (1), 

sin.  19°  57'  =9.533009 

sec.  40°  42'  =0.120254 

Amplitude  =  E.  26°  44'  49"  N.  sin.  =9.653263 
or  Azimuth    =N.  63°  15'  llx/  E. 

By  formula  (2), 

tang.  19°  57'  =9.559885 

tang.  40°  42'  =9.934567 

EM  =  18°  11'  34-       sin.  =9.494452 
Hence  the  hour  angle  =  7h.  12m.  46.3s. 

Ex.  2.  Find  the  hour  angle  and  amplitude  of  Antares  (Dec. 
26°  6'  S.),  when  it  sets  to  an  observer  at  Philadelphia,  Lat.  39° 
57'  N.  Am.  Hour  angle =4h.  23m.  5.7s. 

Amplitude  =W.  35°    V  16"  S. 
or  Azimuth     =S.    54°  58'  44"  W. 

As  we  have  frequent  occasion  to  know  the  time  of  rising  and 
setting  of  the  heavenly  bodies,  it  is  convenient  to  have  a  table 
from  which  this  may  be  ascertained  without  the  labor  of  com- 
putation. Table  XIX.  furnishes  the  semi-diurnal  arcs  for  any 
latitude  up  to  60°,  and  for  any  declination  not  exceeding  29°, 
from  which,  if  we  know  the  time  of  passing  the  meridian,  the 
time  of  rising  or  setting  is  easily  found. 

To  find  the  time  of  rising  of  the  sun's  upper  limb,  corrected 
for  refraction,  see  Art.  169. 

RING    MICROMETER. 

(150.)  The  ring  micrometer 
consists  of  an  opaque  ring,  in- 
serted in  the  focus  of  a  telescope, 
and  having  a  diameter  somewhat 
less  than  that  of  the  field  of  view. 
When  the  telescope  is  fixed  in  po- 
sition, by  observing  the  instants 
at  which  two  stars  pass  the  op- 
posite sides  of  either  the  outer  or 
inner  circle  of  the  ring,  their  dif- 
ference of  right  ascension  and 


116 


PRACTICAL    ASTRONOMY. 


declination  may  be  computed, 
provided  we  know  the  diameter 
of  the  ring.  The  annexed  fig- 
ure represents  the  appearance  of 
a  ring  suspended  in  the  focus  of 
a  telescope,  the  field  of  view  be- 
ing represented  by  the  circle 
NWSE.  Each  star  is  to  be  ob- 
served when  it  passes  behind  the 
ring  at  L,  when  it  reappears  at 
A ;  when  it  disappears  again  at 
A7,  and  when  it  reappears  at  M. 

(151.)  To  determine  the  radius  of  the  ring. 
If  there  are  spider  lines  bisecting  the  ring  exactly  in  the  cen- 
tre, we  may  determine  the  radius  by  observing  the  time  required 
by  an  equatorial  star  in  passing  centrally  across  the  ring  ;  or  by 
observing  the  passage  of  any  star  not  very  near  the  pole,  and 
multiplying  the  interval  by  the  cosine  of  its  declination ;  that  is, 

r— Radius  =  — (£x  —  £)cos.  Dec., 
£ 

where  t  and  V  are  the  times  of  ingress  and  egress  of  the  star. 
See  Art.  72. 

The  radius  of  the  ring  will  retain  the  same  value  only  as  long 
as  the  distance  of  the  ring  from  the  object-glass  remains  un- 
changed. "When,  therefore,  the  radius  of  the  ring  has  been  once 
determined,  the  position  of  the  tube  carrying  the  micrometer 
should  be  accurately  marked,  and,  in  all  subsequent  observa- 
tions, should  be  carefully  adjusted  to  the  same  position. 

(152.)  To  determine  the  difference  of  right  ascension  of  two 
stars. 

Point  the  telescope  in  such  a  manner  that  the  stars  may  trav- 
erse the  ring  in  succession ;  one  of  them,  for  example,  from  A  to 
Ax,  the  other  from  B  to  Bx,  and  leave  the  telescope  undisturbed 
during  the  observation.  Note  the  times  T  and  Tx,  corresponding 
to  the  instants  of  ingress  and  egress  of  the  first  star  at  A  and  Ax ; 
again,  leaving  the  telescope  undisturbed,  note  the  times  t  and  t', 
corresponding  to  the  instants  of  ingress  and  egress  of  the  second 
star  at  B  and  B'.  The  instant  of  passing  the  middle  point,  D, 
of  the  chord  A  A',  will  be  denoted  by  tJ(T'-f-T) ;  and  the  instant 


THE    DIURNAL    MOTION. 


117 


of  passing  the  middle  point,  H,  of  the  chord  BBX,  will  be  de- 
noted by  i  (£'  +  £)•  The  difference  of  right  ascension  will  there- 
fore be  £(t'+t)  —  i(T'-f  T),  provided  the  clock  has  no  rate  that 
sensibly  affects  the  interval. 

(153.)  To  determine  the  difference  of  declination  of  two  stars. 

"We  must  previously  have  an  approximate  knowledge  of  the 
declination  of  each  of  the  stars. 

Put  d  =the  approximate  declination  of  the  first  star  ; 
(T=the  approximate  declination  of  the  second  star. 

Then  we  shall  have 

AD  =  J(T'  —  T)15  cos.  d, 
and  BK=%(t'-t)l5  cos.  6'. 

Put       X=the  angle  ACD,  and  x=the  angle  BCH. 


Then 


Also, 


sin.  X=—  =^  cos.  d  (T'-T) 
r      2r 


sn. 


=  cos.  6'(t'-t) 

r       2r 


(2) 


CD=:rcos.  X 
CH=r  cos.  x 

Hence  DH,  or  the  difference  of  declination =r(cos.  a;— cos.  X), 
when  both  arcs  are  on  the  same  side  of  the  centre  of  the  ring. 
When  they  are  on  opposite  sides,  the  difference  of  declination 
=  r(cos.  x-f  cos.  X). 

When  the  observations  are  made  with  reference  to  the  outer 
edge  of  the  ring,  we  must  proceed  in  the  same  manner ;  and  if 
observations  are  made  at  both  edges  of  the  ring,  a  mean  of  the 
two  results  must  be  taken. 

The  results  for  right  ascension  will  be  most  reliable  when  the 
stars  pass  near  the  centre  of  the  ring ;  but  the  results  for  decli- 
nation will  be  most  reliable  when  the  stars  pass  at  a  considera- 
ble distance  from  the  centre. 

(154.)  The  following  observations  of  Encke's  comet  and  a 
neighboring  star  will  illustrate  the  use  of  this  micrometer : 


North  or 
South  of 
Centre. 

Outer  Ring. 
Ingress. 

Inner 
Ring. 
Ingress. 

Inner 
Ring. 
Egress. 

Outer 
Ring. 
Egress. 

Concluded 
Transit  over 
Hour-Circle. 

Difference 
ofR.  A. 

Star 
Comet 

N. 
S. 

h.      m.       s 

23  13     9 
23  15  11 

13  28 
15  36 
36  23 
38  15 

772.         S. 

14  44 

16  38 

m.      s. 

15     4 

17     7 

m.           s. 

14     6.25 
16     8.0 

m.        s. 

2     1.75 

Star 
Comet 

N. 
S. 

23  35  59 
23  37  48 

37  16 
39  10 

37  41 
39  37 

36  49.75 
38  42.5 

1  52.75 

118  PRACTICAL   ASTRONOMY. 

The  observations  of  the  first  of  the  preceding  stars  with  the 
outer  ring  give  J(T'  +  T)  =  14m.  6.5s.  ;  from  the  inner  ring  we 
obtain  14m.  6s.  ;  the  mean  of  the  two  is  14m.  6.25s.,  which  is 
the  time  of  passing  the  middle  point  of  its  chord.  In  the  same 
manner  we  obtain  for  the  comet  16m.  8.0s.  Hence  their  differ- 
ence of  right  ascension  was  2m.  1.75s.  ;  and,  in  the  same  man- 
ner, their  difference  of  right  ascension  at  the  second  observation 
was  1m.  52.75s. 

The  difference  of  declination  is  computed  as  follows,  using 
only  the  observations  of  the  inner  ring  : 

The  radius  of  the  inner  ring  was  9X  38"  =578". 

The  declination  of  the  star  was  32°  8'  41"  N. 

The  declination  of  the  comet  at  the  first  observation  was  31° 
56'  25"  N.  nearly. 

The  declination  of  the  comet  at  the  second  observation  was 
31°  53'  14"  N.  nearly. 

For  the  first  star  observation,  by  equation  (1), 
i(*'_f)  =  38s.  =  1.57978 
15    =  1.17609 
cos.  (5=9.92773 

-=7.23807 
r 

x=56°  36'  50"  sin.  =  9.92167 
By  equation  (2), 

cos.  x=  9.74058 
r=2.76193 


=  318".1=2.50251 

In  a  similar  manner,  for  the  first  comet  observation,  we  ob- 
tain 

CD=422".3. 

Hence,  since  the  star  and  comet  were  on  opposite  sides  of  the 
centre,  the  difference  of  decimation  =318".1  +  422".3  =  740".4 
=  12/20//.4. 

In  the  same  manner,  we  find  the  difference  of  declination  at 
the  second  observation  to  be  15'  29".7. 

(155.)  Frequently  a  comet  changes  its  right  ascension  and 
declination  so  rapidly  that  we  can  not  assume  that  in  one  sec- 
ond of  time  it  describes  15"  cos.  d  in  arc,  and  that  its  path  is 
perpendicular  to  an  hour  circle.  In  this  case,  we  must  apply  a 


THE   DIURNAL   MOTION.  119 

correction  to  the  result  obtained  without  regarding  the  proper 
motion. 

Let  NS  represent  an  hour  circle,  and  draw  BB'  perpendicular 
toNS. 

Suppose  the  comet  to  describe  the  path 
BK  instead  of  BB7, 

Represent  CGr  by  6?= the  least  distance 
of  the  comet  from  the  centre  of  the  ring ; 
and  let  r=£  (f—t)=ba&  the  interval  be- 
tween the  ingress  and  egress ;  then 
d?=r2 —  (15T  cos.  d)2. 

Represent  by  Aa  the  increase  of  right  "~~s~ 

ascension  of  the  comet  hi  a  second  of  time ;  AT  the  change  of 
T  caused  by  the  change  of  right  ascension,  so  that  T+AT  repre- 
sents the  half  interval  which  would  have  been  observed  if  there 
had  been  no  change  of  right  ascension.  Then 


But,  by  differentiating  the  above  expression  for  d2,  we  have 

,        152rcos.2<$ 

Aa=  --  -  -  Ar. 
a 

Hence  Arf=(15r  cos.  6)^  .......  (A) 

ct 

which  represents  the  required  correction  of  the  comet's  declina- 
tion. 

Let  A(5  represent  the  change  of  declination  of  the  comet  in  a 
second  of  time,  and  n  the  angle  KBBX,  which  the  comet's  path 
makes  with  a  parallel,  we  shall  have 


or  we  may  assume  without  appreciable  error, 

Ad 

tang.  n=—  -  . 
lo  cos.  6 

Let  y  represent  GI,  the  portion  of  the  comet's  path  between 
the  hour  circle,  CI,  and  the  perpendicular,  CGr,  drawn  from  the 
centre  upon  the  path,  and  we  shall  have 

c?A(5 

77—  a  tang.  n=  -  . 
15  cos.  6 

The  correction  to  be  applied  to  the  time  of  transit  over  the 
hour  circle,  determined  without  regard  to  proper  motion,  is 


120  PRACTICAL   ASTRONOMY. 


15  cos.  6' 

-= 


In  the  example  given  above,  the  comet's  motion  in  right  as- 
cension in  24  hours  was  —  7m.59.25s.,  and  in  declination  —3° 
5'  Ox/.7.  Consequently, 

log.  Aa=7.74405ra, 

and  log.  Ad=9.10884rc. 

Moreover,  we  have  before  found, 

lo.  d=  2.62567 


<J=31056'25". 

To  compute  Ad.  To  compute  AT. 

By  formula  A.  By  formula  B. 

15  =  1.17609  d=  2.62567 

T=31s.  =  1.49136  &d=  9.10884^ 

cos.  (5  =  9.92870  1.73451» 

2.59615  (15  cos.  d)2=  2.20958 


2  Ar=  -  0.33s.  =  9.52493ra. 


5.19230 
A<z  =  7.74405;* 
comp.  d=  7.37433 
Arf=-2".04= 0.31068ft. 

At  the  time  of  these  observations,  the  comet  was  moving 
southward  from  the  centre  of  the  field,  so  that  its  apparent  path 
may  be  represented  by  BK.  It  passed  the  point  Gr,  half-way 
between  B  and  K,  at  16m.  8.0s.  Hence  it  passed  the  point  I,  on 
the  hour  circle  bisecting  the  ring,  at  16m.  8.33s.  Therefore  the 
true  difference  of  right  ascension  was  2m.  2.08s. 

The  comet's  least  distance  from  the  centre  of  the  ring  was  be- 
fore computed  to  be  422".3,  which  is  represented  by  CGr.  Its 
corrected  distance  is  CH,  which  is  420".3.  Hence  the  true  dif- 
ference of  declination  was 

420".3  +  318x/.l  =  738".4 = 12X  18/x.4. 


CHAPTER  V. 

TIME. 

(156.)  THE  interval  between  two  successive  returns  of  the 
vernal  equinox  to  the  same  meridian  is  called  a  sidereal  day. 

The  interval  between  two  successive  returns  of  the  sun  to  the 
same  meridian  is  called  a  solar  day. 

The  sun  completes  an  apparent  revolution  about  the  earth 
in  one  year,  or  365  days  5  hours  48  minutes  and  47.57  sec- 
onds ;  so  that  the  sun's  mean  daily  motion  is  found  by  the  pro- 
portion 

one  year :  one  day ::  360°  :  daily  motion =59/  8/x.33. 

This  motion  is  not  uniform,  but  is  greatest  when  the  sun  is 
nearest  the  earth.  Hence  the  solar  days  are  unequal ;  and  to 
avoid  the  inconvenience  which  would  result  from  this  fact,  as- 
tronomers have  recourse  to  a  mean  solar  day,  the  length  of  which 
is  equal  to  the  mean  or  average  of  all  the  apparent  solar  days 
in  a  year. 

(157.)  The  length  of  the  mean  solar  day  is  different  from  that 
of  the  sidereal,  because  when  the  mean  sun,  in  its  diurnal  mo- 
tion, returns  to  the  meridian,  it  is  59X  8/x.33  advanced  eastward 
in  right  ascension. 

An  arc  of  the  equator,  equal  to  360°  59'  8".33,  passes  the 
meridian  in  a  mean  solar  day,  while  only  360°  pass  in  a  sidereal 
day.  To  find  the  excess  of  the  solar  day  above  the  sidereal  day, 
expressed  in  sidereal  time,  we  have  the  proportion 

360°  :  59'  8/x.33  ::  one  day :  3m.  56.555s. 

Hence  24  hours  of  mean  solar  time  are  equivalent  to  24h. 
3m.  56.555s.  of  sidereal  time.  As  we  have  frequent  occasion 
to  convert  intervals  of  mean  solar  time  into  intervals  of  sidereal 
time,  Table  IV.  has  been  constructed,  from  which  such  inter- 
vals are  found  by  mere  inspection. 

Example.  Find  the  sidereal  interval  which  corresponds  to  I5h. 
20m.  20.58s.  of  mean  solar  time. 


122  PRACTICAL   ASTRONOMY. 

According  to  Table  IV., 

15  hours  mean  solar  time= 15h.    2m.  27.847s.  sidereal  time. 
20  minutes"       «      "     =         20        3.285         "         « 
20  seconds   "       "      "     =  20.055         "          " 
0.58             "       "      "    =  0.582         "         " 
The  sidereal  interval        =15h.  22m.  51.769s. 

To  find  the  excess  of  the  solar  day  above  the  sidereal  day,  ex- 
pressed in  solar  time,  we  have  the  proportion 

360°  59'  8x/.33  : 59'  8".33  ::  one  day :  3m.  55.909s. 

Hence  24  hours  of  sidereal  time  are  equivalent  to  23h.  56m. 
4.091s.  of  mean  solar  time.  In  order  to  facilitate  the  conver- 
sion of  sidereal  time  into  solar  time,  Table  V.  has  been  construct- 
ed, from  which  these  intervals  are  found  by  mere  inspection. 

Example.  Find  the  solar  interval  which  corresponds  to  16h. 
15m.  25.66s.  of  sidereal  time. 

According  to  Table  V., 

16  hours  sidereal  time  =  15h.  57m.  22.727s.  mean  solar  time. 
15  minutes  "         "     =         14      57.543       "        "        " 
25  seconds    "         "     =  24.932       "        "        " 
0.66             "         "     =  0.658       "        "        " 
The  solar  interval        =16h.  12m.  45.860s. 

(158.)  Throughout  this  work  we  shall  suppose  the  student  to 
have  in  his  possession  some  astronomical  ephemeris,  like  the 
Nautical  Almanac.  The  English  Nautical  Almanac  has  been 
published  annually  since  1767,  and  generally  appears  about 
three  years  in  advance  of  the  date  for  which  it  is  computed. 
The  French  Connaissance  des  Temps  has  been  published  annu- 
ally since  1679,  without  ever  having  suffered  a  single  interrup- 
tion; and  the  Berlin  Astronomisches  Jahrbuch  has  been  pub- 
lished annually  since  1776.  The  first  volume  of  the  American 
Nautical  Almanac,  being  for  1855,  was  published  in  February, 
1853,  and  is  expected  to  appear  regularly  in  future.  Either  of 
these  almanacs  will  furnish  all  the  data  which  are  required  for 
the  computations  in  this  treatise.  We  shall,  however,  employ 
the  American  Nautical  Almanac  for  1855  or  1856,  whenever  it 
can  conveniently  be  done  ;  and  for  other  cases  shall  refer  to  the 
English  Nautical  Almanac. 


TIME.  123 

%       PROBLEM. 

(159.)   To  convert  mean  solar  time  into  sidereal  time. 

"When  the  sun  is  on  the  meridian,  the  sidereal  time  is  the  same 
as  the  sun's  apparent  right  ascension. 

Thus,  according  to  the  American  Nautical  Almanac  for  1855, 
page  271,  the  sun's  apparent  right  ascension  at  Washington,  ap- 
parent noon,  January  1,  1855,  is  18h.  46m.  58.21s. ;  this  is, 
therefore,  the  sidereal  time  at  that  instant.  The  sidereal  time 
of  mean  noon  may  be  found  from  the  preceding,  by  applying  the 
equation  of  time,  reduced  to  its  sidereal  equivalent.  Thus,  on 
January  1, 1855,  the  equation  of  time  is  +  3m.  49.72s.,  which  is 
equivalent  to  3m.  50.35s.  sidereal  time.  Therefore  the  sidereal 
time  of  mean  noon  is 

18h.  46m.  58.21s.  -3m.  50.35s.,  which  equals  18h.  43m.  7.86s. ; 
and  this  is  the  number  given  in  the  last  column  of  page  271  of 
the  almanac.  The  almanac  furnishes,  in  like  manner,  the  side- 
real time  of  mean  noon  at  Washington  for  every  day  in  the  year. 
With  this  assistance,  we  can  easily  convert  any  instant  of  mean 
solar  time  into  its  corresponding  sidereal  time,  by  the  following 

RULE. 

Sidereal  time  required= sidereal  time  at  the  preceding  mean 
noon,  plus  the  sidereal  interval  corresponding  to  the  given 
mean  time. 

Example.   Convert   2h.  22m.  25.62s.  mean   solar   time   at 
Washington,  January  2,  1855,  into  sidereal  time. 
Sidereal  time   at  the  preceding  mean 

noon,  viz.,  January  2 18h.  47m.    4.42s. 

Add  the  mean  time,  reduced  to  its  side- 
real equivalent  by  Table  IV 2h.  22m.  49.02s. 

The  sum  is  the  sidereal  time  required  21h.    9m.  53.44s. 
If  the  place  of  observation  be  not  on  the  meridian  of  the  ephem- 
eris,  the  sidereal  time  at  mean  noon  must  be  corrected  by  the 

addition  of  9.8565s.  (  ——  '  *  -J  for  each  hour  of  longi- 
tude, if  the  place  be  to  the  west  of  the  first  meridian,  but  by  its 
substr action  if  to  the  east. 

Example.  Convert  7h.  55m.  51.65s,  mean  time  at  the  High 


124  PRACTICAL   ASTRONOMY. 

School  Observatory,  Philadelphia,  April  19,  1855,  into  sidereal 
time. 

The  sidereal  time  at  the  preceding  Wash- 
ington mean  noon  is  .  .  .^U.; Ih.  48m.  55.82s. 

Correction  for   7m.  33.6s.,  Philadelphia 

east  of  Washington —1.24s. 

Sidereal  time  at  the  preceding  Philadel- 
phia mean  noon Ih.  48m.  54.58s. 

Add  the  mean  time,  reduced  to  its  side- 
real equivalent 7h.  57m.    9.82s. 

The  sum  is  the  sidereal  time  required  .  .  9h.  46m.    4.40s. 

PROBLEM. 

(160.)  To  convert  sidereal  time  into  mean  solar  time. 

If  from  the  proposed  sidereal  time  we  subtract  the  sidereal 
time  at  the  preceding  mean  noon,  we  shall  obtain  the  interval 
from  mean  noon  expressed  in  sidereal  time ;  and  if  we  convert 
this  interval  into  its  mean  solar  equivalent,  we  shall  have  the  in- 
terval elapsed  since  mean  noon  expressed  in  mean  time,  and  there- 
fore the  time  which  ought  to  be  shown  by  a  mean-time  clock. 

Example.  Convert  21h.  9m.  53.44s.  sidereal  time  at  Wash- 
ington, January  2,  1855,  into  mean  solar  time. 

Sidereal  time  given 21h.    9m.  53.44s. 

Sidereal  time  at  preceding  mean  noon  .  18h.  47m.    4.42s. 

Interval  in  sidereal  time  from  mean  noon     2h.  22m.  49.02s. 

Equivalent  in  mean  solar  time  by  Ta- 
ble Y tefcrttfHf 2h.  22m.  25.62s. 

which  is  therefore  the  mean  solar  time  required. 

(161.)  If  we  subtract  the  sidereal  time  at  mean  noon  from 
twenty-four  hours,  and  convert  this  interval  into  its  solar  equiv- 
alent, we  shall  have  the  mean  time  of  transit  of  the  first  point 
of  Aries,  which  may  be  called  the  mean  time  at  sidereal  noon. 
It  is  the  time  which  ought  to  be  shown  by  a  mean-time  clock, 
at  the  moment  that  a  clock  adjusted  to  sidereal  time  indicates 
exactly  Oh.  Om.  Os.  The  mean  time  of  transit  of  the  first  point 
of  Aries  is  given  in  the  English  Nautical  Almanac  for  every  day 
of  the  year,  on  page  xx.  of  each  month.  It  is  omitted  in  the 
American  Almanac  for  1855,  but  is  inserted  in  the  Almanac  for 
1856,  on  page  in.  of  each  month,  under  the  title  mean  time  of 
sidereal  Oh.  This  quantity  is  found  as  follows : 


TIME.  125 

The  sidereal  time  at  Greenwich,  mean  noon,  January  1, 1855, 
is  18h.  42m.  17.25s.  Subtracting  this  from  24  hours,  we  have 
5h.  17m.  42.75s.,  which,  reduced  to  its  equivalent  solar  interval, 
is  5h.  16m.  50.70s.,  which  is,  therefore,  the  mean  time  of  transit 
of  the  first  point  of  Aries  for  January  1, 1855,  at  Greenwich,  and 
is  so  given  on  page  xx.  of  the  English  Almanac.  "With  this  as- 
sistance, we  can  easily  convert  any  instant  of  sidereal  time  into 
its  corresponding  mean  solar  time,  by  the  following 

RULE. 

The  mean  solar  time  required— mean  time  at  the  preceding 
sidereal  noon,  plus  the  mean  interval  corresponding  to  the 
given  sidereal  time. 

Example.  Convert  21h.  8m.  55.39s.  sidereal  time  at  Green- 
wich, January  2,  1855,  into  mean  time. 
Mean  time  at  the  preceding  sidereal  noon, 

January  1 5h.  16m.  50.70s. 

Add  the  given  sidereal  time  reduced  to 

its  equivalent  mean  time 21h.    5m.  27.51s. 

The  sum  is  the  mean  time  required,  Jan- 
uary 2 2h.  22m.  18.21s. 

(162.)  If  the  place  of  observation  be  not  on  the  meridian  of 
the  ephemeris,  the  mean  time  of  the  transit  of  the  first  point 
of  Aries  must  be  corrected  by  the  subtraction  of  9.8296s. 

/     3m.  55.909sA 

I  = — —    -  I  for  each  hour  of  longitude,  if  the  place  be  to 

the  west  of  the  first  meridian,  but  by  its  addition  if  to  the  east. 
Example.  Convert  22h.  llm.  37.68s.  sidereal  time  at  High 
School  Observatory,  Philadelphia,  October  17,  1855,  into  mean 
time. 

The  mean  time  at  the  preceding  Green- 
wich sidereal  noon  is lOh.  20m.  32.74s. 

Correction  for  5h.  Om.  37.6s.,  Philadel- 
phia west  of  Greenwich — 49.25s. 

Mean  time  at  the  preceding  Philadelphia 

sidereal  noon lOh.  19m.  43.49s. 

Add  the  sidereal  time,  reduced  to  its 

mean  equivalent 22h.    7m.  59.52s. 

The  sum  is  the  mean  time  required   .  .     8h.  27m.  43.01s. 


126 


PRACTICAL   ASTRONOMY. 


PROBLEM. 

(163.)  To  find  the  time  by  observation. 

First  Method. — By  equal  altitudes  of  a  star  on  opposite 
sides  of  the  meridian. 

Observe  the  times  when  the  star  has  equal  altitudes  before 
and  after  passing  the  meridian ;  the  arithmetical  mean  between 
these  times  is  the  time  of  the  star's  passing  the  meridian.  By 
comparing  this  time  with  the  known  place  of  the  star,  we  may 
obtain  the  error  of  the  clock. 

Example.  The  numbers  in  column  first  of  the  following  table 
show  the  times  when  Arcturus  had  the  altitudes  contained  in 
column  second,  on  the  east  of  the  meridian.  Column  third  shows 
the  times  when  it  had  the  same  altitudes  on  the  west  of  the  me- 
ridian. Column  fourth  shows  the  sums  of  these  times,  the  av- 
erage of  which  is  28h.  7m.  42.5s. ;  consequently  the  star  passed 
the  meridian  at  14h.  3m.  51.25s.  by  the  clock. 


East. 

Altitude. 

West. 

Sum. 

h.      m.         s. 

10  55  49.2 
57  59.5 
11     0     9.7 
2  20.7 
6  43.7 

43  10 
43  30 
43  50 
44  10 
44  50 

A.      m.        s. 

17  11  53.0 
9  43.0 
7  32.5 
5  22.2 
0  59.0 

h.    m.        s. 

28  7  42.2 
42.5 
42.2 
42.9 
42.7 

Mean      ....         28  7  42  5 

Meridian  passage  =  14  3  51.25 

If  we  suppose  the  clock  regulated  to  sidereal  time,  and  the 
right  ascension  of  the  star  to  be  14h.  9m.  0.16s.,  then  the  clock 
was  slow  5m.  8.91s. 

(164.)   Second  Method. — By  equal  altitudes  of  the  sun. 

Since  the  declination  of  the  sun  changes  from  morning  to 
evening,  the  time  of  the  sun's  arriving  at  a  given  altitude  is 
affected  by  this  motion,  and  we  must  compute  the  correction 
to  be  applied  to  the  mean  of  the  times  observed.  This  may  be 
done  by  the  following  method : 

Let  PZM  be  the  meridian  of  the  place  of  observation,  P  the 
pole,  Z  the  zenith,  AMB  a  small  circle  parallel  to  the  horizon, 
ANB  the  parallel  described  by  a  star  in  its  diurnal  motion,  and 
cutting  the  former  circle  in  A  and  B.  If  ZA  is  found  by  obser- 


TIME. 


127 


vation  equal  to  ZB,  then,  since  PZ  is  constant,  if  the  polar  dis- 
tance, PA,  does  not  change,  the  two  triangles,  PZA,  PZB,  will  be 
mutually  equilateral,  and,  consequently,  the  angle  ZPA  = 
that  is,  the  hour  angle  of  a  star 
from  the  meridian  is  the  same 
for  equal  altitudes  on  the  east 
and  west  sides  of  the  meridian ; 
and  this  is  the  case  with  all 
the  fixed  stars,  but  not  with 
the  sun.  Suppose  the  polar 
distance  of  the  sun  has  di- 
minished during  the  interval, 
then,  when  the  western  hour 
angle,  ZPB,  is  equal  to  the  east- 
ern, ZPA,  the  sun  will  be  at  Bx, 
nearer  to  the  zenith ;  and  when 
the  sun  reaches  the  circle  AMB  at  C,  the  hour  angle  ZPC  will 
be  greater  than  the  hour  angle  ZPA  or  ZPB. 

It  is  necessary,  then,  to  compute  the  angle  BPC. 

Put  0=the  latitude  of  the  place;  6= the  declination  of  the 
sun  when  on  the  meridian ;  dd— the  increase  of  declination  from 
the  meridian  to  the  afternoon  observation;  P=the  hour  angle 
from  the  meridian,  supposing  no  change  in  the  declination; 
dP  —  thQ  increase  of  the  hour  angle  in  time  caused  by  the  change 
of  declination ;  and  Z^the  observed  zenith  distance. 

Now,  in  the  triangle  APZ,  Trig.,  Art.  2^5, 

cos.  AZ=rcos.  PZ  cos.  AP  +  sin.  PZ  sin.  AP  cos.  APZ, 
or        cos.  Z^sin.  0  sin.  d-f-cos.  0  cos.  6  cos.  P (1) 

Also,  in  the  triangle  CPZ, 

cos.  CZ=cos.  PZ  cos.  CP  +  sin.  PZ  sin.  CP  cos.  CPZ, 
or     cos.  Z  — sin.  </>  sin.  (6+ dd)  +  cos.  0  cos.  (d+dd)  cos.(P  +  dP) 
—sin.  0  sin.  6  cos.  dd+sin.  0  cos.  d  sin.  dd 
+  cos.  $  (cos.  6  cos.  dd— sin.  6  sin.  dd)  (cos.  P  cos.  dP 
-sin.  P  sin.  dP). 

But  since  the  variations  of  6  and  P,  in  the  present  case,  are 
necessarily  small,  we  may  put 
cos.6to=l;  cos.dP  =  l;  sin.rf(J=d<Jsin.l//;  sin.  dP  =  15dP  sin.  V. 

Therefore, 


128  PRACTICAL   ASTRONOMY. 

cos.  Z  =  sin.  0  sin.  d  +  dd  sin.  1"  sin.  </>  cos.  d+cos.  <£  cos.  d  cos.  P 

—  dd  sin.  1"  cos.  <f>  sin.  d  cos.  P 

—  15dP  sin.  1"  cos.  <j>  cos.  d  sin.  P. 
Hence,  by  equation  (1), 

0  =  dd  sin.  </>  cos.  d — dd  cos.  0  sin.  d  cos.  P 

—  15dP  cos.  $  cos.  d  sin.  P. 
Whence 

*"  sin.  d)  cos.  d— dd  cos.  d>  sin.  d  cos.  P 


dP  = 


15  cos.  $  cos.  d  sin.  P 


or  dP  =  r(tang.  0  cosec.  P  — tang,  d  cot.  P) (2) 

15 

which  is  the  correction  to  be  applied  to  the  mean  of  the  times 
observed. 

If  the  sun's  motion  in  declination  is  northward,  this  correction 
is  to  be  subtracted  from  the  mean  of  the  times  observed ;  if  the 
motion  is  southward,  it  must  be  added. 

Ex.  1.  At  a  place  in  Lat.  54°  20'  N.,  the  sun  was  found  to 
have  equal  altitudes  at  8h.  59m.  4s.  A.M.  and  at  3h.  Om.  40s.  P.M. 
It  is  required  to  find  the  time  of  noon,  the  declination  of  the  sun 
being  19°  48'  29"  N.,  and  the  decrease  of  declination  between 
the  two  observations  being  1927'. 

By  equation  (2), 

tang.  </>= 0.14406  tang.  6= 9.55652 

cosec.  P=0.14900  cot.  P  =  9.99697 

^=6.4=0.8Q618  ^=0.80618 

15  15 

12.57s.  =  1.09924  2.29s.  =  0.35967  "*o 

Hence  12.57  -  2.29  =  10.28s. = dP. 

This  correction  is  to  be  added  to  the  mean  of  the  times  ob- 
served, because  the  sun's  motion  was  southward. 

The  mean  of  the  observed  times  is  llh.  59m.  52s. ;  therefore 
the  time  of  apparent  noon  was  Oh.  Om.  2.28s.,  or  the  clock  was 
2.28s.  too  fast  by  apparent  time. 

(165.)  In  order  to  facilitate  the  preceding  computations,  vari- 
ous tables  have  been  devised,  but  the  one  which  has  been  chiefly 
used  was  first  proposed  by  Grauss.  Table  XL  is  from  the  Amer- 
ican Nautical  Almanac  for  1856,  and  was  furnished  by  Professor 
Chauvenet.  It  differs  from  the  table  of  Gauss  only  in  using  the 


$  TIME.  129 

hourly  change  of  the  sun's  declination  instead  of  twice  the  daily 
change.     This  table  was  constructed  as  follows  : 

Put  T  =  the  interval  of  time  between  the  morning  and  after- 

noon observations,  expressed  in  hours. 
p  =  ihe  hourly  change  of  the  sun's  decimation. 
Then,  since  dd  represents  the  increase  of  declination  from  the 
meridian  to  the  afternoon  observation,  we  shall  have 


And  since  P  represents  the  hour  angle  from  the  meridian  ex- 
pressed in  arc,  we  shall  have 


Hence  the  correction  to  be  added  to  the  mean  of  the  times 
observed  to  obtain  the  time  of  apparent  noon  is 
_pTtang.  0        ^T  tang,  d 
30  sin.  7JT     30  tang.  7JT' 


or        *  =  -,  tang.  ^+P  tang.  ^ 
Let  us  make 


~3Tsin.  7£T'  ~30langT7iT' 

and  we  shall  have 

x=—Ap,  tang.  (f> -\-~Bfi,  tang.  6. 

Table  XL  furnishes  the  values  of  A  and  B  for  all  values  of  T 
from  2  hours  to  24  hours.     The  following  is  the  method  of  com- 
puting A  and  B : 
Let  T  =  2  hours. 

7iT  =  15°  sin.=9.4130  7iT  =  15°  tang.  =  9.4280 

30=1.4771  30=1.4771 

0.8901  0.9051 

2  =  0.3010  2  =  0.3010 

log.  A = 9.4109  log.  B  =  9.3959 

which  are  the  values  of  log.  A  and  log.  B,  given  in  the  table  for 
an  interval  of  2  hours ;  and  in  the  same  manner  were  the  other 
numbers  computed.  If  we  employ  the  numbers  of  this  table, 
the  computation  of  Ex.  1  will  proceed  as  follows  : 

The  interval  of  time  between  the  morning  and  afternoon  ob- 
servations being  6h.  1m.  36s.,  we  have,  by  Table  XL,  log.  A= 
9.4520,  and  log.  B= 9.2999;  and  by  the  Nautical  Almanac, 
fj,=  —  31".85.  The  operation,  therefore,  will  stand  thus  : 

I 


130 


PRACTICAL   ASTRONOMY. 


log.  A  =  9.4520  log.  B  =  9.2999 

p=_31.85  =  1.5031»  p= -31.85  =  1.5031* 

tang.  0  =  0.1441  tang.  (5=9.5565 

-  12.57s.  =  1.0992»  -  2.29s.  =  0.3595^ 

Hence  x  =  12.57  —  2.29  =  10.28,  the  same  correction  as  found 
on  page  128. 

The  following  rule  for  the  signs  of  the  two  terms  of  the  cor- 
rection for  equal  altitudes  may  be  found  convenient : 

The  sign  of  the  first  term  is  positive  from  the  summer  to 'the 
winter  solstice,  and  negative  from  the  winter  to  the  summer 
solstice. 

The  sign  of  the  second  term  is  positive  from  the  equinoxes  to 
the  solstices,  and  negative  from  the  solstices  to  the  equinoxes. 

(166.)  The  following  is  the  most  convenient  mode  of  taking 
these  observations.  Having  brought  the  lower  limb  of  the  sun, 
as  seen  reflected  from  the  sextant  mirror,  into  approximate  con- 
tact with  the  upper  limb,  as  seen  reflected  from  the  mercury, 
move  the  vernier  forward,  and  set  the  zero  to  coincide  with  some 
convenient  division  upon  the  limb.  Wait  for  the  instant  of  con- 
tact, and  note  the  time  by  the  chronometer.  Move  the  vernier 
forward  10'  or  20',  and  note  the  instant  of  contact  as  before, 
making  the  successive  observations  at  equal  intervals  of  10'  or 
20'.  It  is  by  no  means  necessary  that  the  sextant  should  indi- 
cate the  true  altitude  of  the  body,  for  it  is  the  peculiar  excel- 
lence of  this  method  that  it  merely  requires  the  observations  to 
be  made  at  the  same  altitude  on  both  sides  of  the  meridian. 

Ex.  2.  At  Pembina,  in  Lat.  48°  58'  34",  the  following  double 
altitudes  of  the  sun's  upper  limb  were  observed  August  22d, 
1849: 


A.M. 

Double  Altitudes.                                 P.M. 

h.  m.      s. 

0         '         "                              h.     m.       s. 

9  0  50 

78  36  45 

2  33  55 

9  1  55 

78  53  45 

2  32  47 

9  3  14 

79  14  15 

2  31  35 

9  4  25 

79  32  30 

2  30  28 

959 

79  43  45 

2  29  23 

968 

80     2  15 

2  28  40 

9  6  48 

80  10  45 

2  28     2 

9  8  12 

80  34  45 

2  26  40 

It  is  required  to  find  the  error  of  the  chronometer,  the  declk 


TIME. 


131 


nation  of  the  sun  being  11°  39'  44"  N.,  and  the  hourly  decrease 
of  declination  being  50/x.77. 

When  we  have  a  number  of  observations  made  at  short  inter- 
vals of  time,  as  in  the  present  instance,  it  is  most  convenient  to 
take  the  average  of  all  the  morning  observations,  which  in  the 
present  case  is  9h.  4m.  35.1s. ;  and  also  the  average  of  the  even- 
ing observations,  which  in  the  present  case  is  2h.  30m.  11.2s., 
and  regard  them  as  constituting  one  complete  observation.  The 
mean  of  these  times  is  llh.  47m.  23.15s. ;  the  correction  of  the 
hour  angle  is  found  to  be  13.98s.  Therefore  the  time  of  appar- 
ent noon  was  llh.  47m.  37.13s.,  or  the  chronometer  was  slow 
by  apparent  time  12m.  22.87s. 

Ex.  3.  It  is  required  to  find  the  error  of  the  chronometer  from 
the  following  observations  of  the  sun's  lower  limb,  made  October 
8th,  1852,  in  Lat.  30°  4X  N. ;  the  sun's  declination  at  noon,  Oc- 
tober 8th,  being  6°  7X  S.,  and  decreasing-57//.17  per  hour. 


A.M. 

Double  Altitudes. 

P.M. 

h.       m.      s. 

0                  ' 

h.    m.       s. 

21     7  27 

73     0 

2  33  59 

8  24 

20 

33     3 

9  23 

40 

32     5 

10  18 

74     0 

31     9 

11  16 

20 

30  12 

12  11 

40 

29  14 

13  11 

75     0 

28  13 

14     9 

20 

27  15 

15  10 

40 

26  15 

16     6 

76     0 

25  20 

Ans.  The  mean  of  the  observed  times  is  23h.  50m.  43.0s. ;  the 
correction  of  the  hour  angle  is  -f-  10.45s.  Hence  the  time  of  ap- 
parent noon  was  23h.  50m.  53.45s. ;  and  since  the  equation  of 
time  was  —12m.  34.77s.,  the  chronometer  was  3m.  28.22s.  too 
fast  by  mean  time. 

(167.)  It  frequently  happens  that  clouds  prevent  our  taking 
the  afternoon  observations  corresponding  to  the  morning  observa- 
tions ;  but  if  the  clouds  subsequently  disperse,  we  may  still  take 
a  series  of  western  altitudes,  and  wait  about  18  hours  to  observe 
the  corresponding  eastern  altitudes.  If  the  observations  are 
made  upon  a  star,  the  mean  of  the  observed  times  will  give  the 
time  of  passage  over  the  lower  meridian.  If  the  observations 


132 


PRACTICAL   ASTRONOMY. 


are  made  upon  the  sun,  the  correction  to  the  mean  of  the  ob- 
served times  will  still  be  given  by  formula  (2),  page  128.  If 
the  sun  is  moving  northward,  it  will  be  further  from  the  upper 
meridian  at  the  time  of  the  eastern  observation  than  at  the  time 
of  the  western,  that  is,  it  will  be  nearer  to  the  lower  meridian. 
Hence  the  correction  given  by  formula  (2)  must  be  added  to  the 
mean  of  the  observed  times  ;  and  if  the  interval  between  the  ob- 
servations exceeds  12  hours,  B  will  be  negative,  because  cot.  P 
will  be  negative.  Hence  the  correction  to  be  added  algebraical- 
ly to  the  mean  of  the  observed  times,  to  obtain  the  time  of  ap- 
parent midnight,  is 

x'—Afj,  tang.  </>  +  B//  tang.  6. 

Ex.  4.  It  is  required  to  find  the  error  of  the  chronometer  from 
the  following  observations  of  the  sun's  lower  limb,  made  Octo- 
ber 8th  and  9th,  1852,  in  latitude  30°  %  N.,  the  sun's  declina- 
tion at  midnight  being  —  6°  19X,  and  decreasing  57//.06  per  hour. 


October  8th,  P.M           !       Double  Altitudes. 

October  9th,  A.M. 

h.     m.      s.               \                ° 

h.      m.      s. 

2  33  59 

73     0 

21     8  43 

33     3 

20 

9  41 

32     5 

40 

10  39 

31     9 

74     0 

11  36 

30  12 

20 

12  34 

29  14 

40 

13  31 

28  13 

75     0 

14  30 

27  15 

20 

15  28 

26  15 

40 

16  28 

25  20 

76     0 

17  26 

Ans.  The  mean  of  the  observed  times  is  llh.  51m.  22.05s. ; 
the  correction  of  the  hour  angle  is  —37.12s.  Hence  the  time 
of  apparent  midnight  was  llh.  50m.  44.93s. ;  and  since  the 
equation  of  time  was  —12m.  42.82s.,  the  chronometer  was  3m. 
27.75s.  too  fast  by  mean  time. 

(168.)  Third  Method. — By  a  single  altitude  of  the  sun  or  a 
star. 

Let  PZH  be  the  meridian  of  the 
place  of  observation,  P  the  pole,  Z 
the  zenith,  and  S  the  place  of  the  sun 
or  star.     If  the  zenith  distance,  SZ, 
"o  has  been  measured  and  corrected  for 


TIME.  133 

refraction,  then  in  the  spherical  triangle,  ZPS,  the  three  sides 
are  known,  viz., 

PZ  =  the  co-latitude  =  -0  ; 
ZS  =the  true  zenith  distance  =z  ; 
PS  =the  north  polar  distance  of  the  sta.r  =  d. 
In  this  triangle  we  can  compute  the  angle  ZPS,  which  is  the 
distance  of  the  star  from  the  meridian. 
By  Trig.,  Art.  226, 


sn.  i- 


sin.  b  sin.  c 

Put  2S  = 


then  sin.  ff  -./sin.  (S-^)  sin.  (S-^) 

v  sin.  i/>  sin.  d 

Ex.  1.  At  a  place  in  Lat.  25°  40'  N.,  the  sun's  correct  cen- 
tral altitude  was  found  to  be  10°  6'  27",  when  his  declination 
was  8°  5/  56"  S.     What  was  his  distance  from  the  meridian  ? 
d=  98°    5'  56"  sin.  (S-V»)  =  9-922746 

z=  79°  53X  33"  sin.  (S-rf)  =  9.593007 

V>  =   64°  20X    0"  cosec.  -0  =  0.045117 

242°  19X  29"  cosec.  d=  0.004353 


S  =  121°    9X44"  2)  9.565223 

S—0  =   56°  49'  44"     i-P  =  37°  18X  53".5  sin.  =9.782612 
S-d=  23°    3X48X/       P  =  74°37/47//, 

or  4h.  58m.  31.1s.  =  apparent  time. 

It  may  be  found  convenient  to  employ  in  our  computation  0, 
the  latitude  of  the  place,  and  d,  the  declination  of  the  star,  rath- 
er than  the  co-latitude  and  polar  distance.  For  this  purpose,  we 
have  only  to  substitute  in  the  preceding  formula  d  for  90°  —  d^ 
and  for  90°  — ,  and. we  shall  obtain 


sm.  *r= 


v- 


xsin. 


cos.  $  cos.  d 

Ex.  2.  At  a  place  in  Lat.  52°  13X  26"  N.,  at  3h.  21m.  13.4s. 
P.M.  by  the  clock,  the  corrected  zenith  distance  of  the  sun's  cen- 
tre was  found  to  be  75°  16X  15",  when  his  declination  was  9° 
33X  30"  S.  Required  the  correction  of  the  clock. 

Ans.  The  true  hour  angle  was  3h.  21m.  22.7s. ;  hence  the 
clock  was  9.3s.  slow. 


134 


PRACTICAL   ASTRONOMY. 


Ex.  3.  On  the  4th  of  March,  1850,  at  13h.  16m.  45.12s.  by 
the  sidereal  clock,  the  zenith  distance  of  a  Lyrse  was  observed 
at  Greenwich  to  be  54°  16'  14/x.58  ;  it  is  required  to  determine 
the  error  of  the  clock,  supposing  the  star's  R.  A.  to  be  18h.  31m. 
50.84s.,  and  its  decimation  38°  38X  39".4  N. 

Ans.  The  clock  was  slow  19.66s. 

Ex.  4.  The  following  double  altitudes  of  the  sun's  upper  limb 
were  observed  August  29, 1849,  in  Lat.  48°  17'  N. : 


Times. 

Double  Altitudes. 

h.      m.     s. 

0               /                /' 

8  58  58 

78     7  45 

9     0  15 

78  27  10 

9     1  14 

78  45  45 

9     2  10 

78  57  30 

935 

79  12  30 

9     4  39 

79  35  45 

9     5  54 

80  54  45 

9     6  41 

80     8  45 

9     7  37 

80  21  15 

9     8  25 

80  37  15 

It  is  required  to  determine  the  error  of  the  chronometer,  the 
sun's  declination  being  9°  16X  20/x  N.,  and  his  semi-diameter 
15'  52".  Ans.  The  chronometer  was  slow  23m.  4.44s. 

The  most  favorable  opportunity  for  determining  the  time  from 
altitudes  of  the  sun  or  a  star  is  when  it  rises  or  falls  most  rap- 
idly. This  happens  when  the  sun  or  star  is  passing  the  prime 
vertical ;  that  is,  when  it  is  nearly  east  or  west.  The  sun's  al- 
titude should  not  be  less  than  10  degrees,  on  account  of  the  ir- 
regular refraction  near  the  horizon.  In  general,  two  or  three 
hours  from  the  meridian  will  be  sufficient. 

(169.)  Corollary.  By  the  same  method  we  may  compute  the 
time  at  which  the  sun's  upper  limb  rises  or  sets,  when  allowance 
is  made  for  refraction.  The  effect  of  refraction  is  to  cause  the 
sun  to  appear  above  the  sensible  horizon  sooner  in  the  morning 
and  later  in  the  afternoon  than  he  actually  is ;  and  when  the 
sun's  upper  limb  coincides  with  the  horizon,  the  centre  is  about 
16'  below.  At  the  instant,  therefore,  of  sunrise  or  sunset,  his 
centre  is  90°  50'  from  the  zenith;  the  semi-diameter  being 
about  16',  and  the  horizontal  refraction  34'. 


TIME.  135 

Ex.  1.  Required  the  time  of  sunset  at  New  York,  Lat.  40° 
42',  at  the  summer  solstice. 

Here          ^=  49°  18X  sin.  (S-^)  =  9.908141 

d=  66°  32'  sin.  (S-d)  =  9.777444 

g=   90°  50'  cosec.  7,6=0.120254 

S  =  103°  20'  cosec.  ^=0.037492 

S—  0=  54°    2'  2)  9.843331 

S-d=  36°48/  P=  56°36'sin.=: 


P  =  113°12'=:7h.  33m. 

Hence  the  sun  sets  at  7h.  33m.  apparent  time  ;  or,  adding 
1m.  for  equation  of  time,  we  have  7h.  34m.  mean  time. 

Ex.  2.  Required  the  mean  time  of  sunset  at  New  Orleans, 
Lat  29°  58X  at  the  winter  solstice  ;  mean  time  being  one  min- 
ute slow  of  apparent  time.  Ans.  5h.  5m. 

(170.)  The  preceding  methods  are  adapted  to  the  use  of  trav- 
elers and  navigators,  as  the  observations  may  all  be  made  with 
a  sextant.  In  fixed  observatories  the  time  is  habitually  found 
by  a  transit  instrument,  which  is  the  most  accurate  method 
known,  as  well  as  the  most  convenient. 

Fourth  Method.  —  To  determine  time  by  the  transit  instru- 
ment. 

The  instant  of  the  sun's  .passing  the  meridian  is  the  time  of 
apparent  noon  ;  and  hence,  if  we  compare  the  sun's  passage  over 
the  meridian  with  a  chronometer,  we  shall  obtain  the  deviation 
of  the  chronometer  from  apparent  solar  time.  If  to  this  we 
apply  the  equation  of  time  with  its  proper  sign,  we  shall  obtain 
the  error  of  the  chronometer  in  mean  time. 

Ex.  1.  The  sun  was  observed  to  pass  the  meridian  at  llh. 
59m.  18.7s.  by  chronometer,  the  equation  of  time  being  +  13m. 
22.5s.  Required  the  error  of  the  chronometer. 

Ans.    Om.  41.3s.  slow  for  apparent  time  ; 
14m.    3.8s.  slow  for  mean  time. 

Ex.  2.  The  sun  was  observed  to  pass  the  meridian  at  llh. 
56m.  12.21s.  by  chronometer  ;  the  equation  of  time  being—  3m. 
56.26s.  Required  the  error  of  the  chronometer. 

Ans.  3m.  47.79s.  slow  for  apparent  time  ; 
Om.    8.47s.  fast  for  mean  time. 

In  a  fixed  observatory  it  is  most  convenient  for  ordinary  pur- 
poses to  employ  sidereal  time.  The  error  of  a  sidereal  clock  or 


136  PRACTICAL   ASTRONOMY. 

chronometer  is  found  in  the  manner  already  explained,  except 
that  we  must  know  the  right  ascension  of  the  object  observed. 
The  right  ascension  of  the  sun  and  100  fixed  stars  is  given  for 
every  day  of  the  year,  in  both  the  English  and  American  Nau- 
tical Almanacs ;  and  the  right  ascension  of  1500  stars  is  given 
in  the  catalogue  at  the  close  of  this  volume. 

Ex.  3.  The  star  Rigel  was  observed  to  pass  the  meridian  of 
Greenwich,  February  6,  1851,  at  oh.  6m.  35.41s.  by  a  sidereal 
clock,  the  star's  right  ascension  being  5h.  7m.  22.97s.  Required 
the  error  of  the  clock.  Ans.  47.56s.  slow. 

Ex.  4.  The  sun's  centre  passed  the  meridian  of  Greenwich, 
May  15, 1851,  at  3h.  25m.  35.17s.  by  the  sidereal  clock,  the  sun's 
right  ascension  being  3h.  26m.  33.78s.  Required  the  error  of 
the  clock.  Ans.  58.61s.  slow. 

(171.)  The  error  of  the  clock  may  be  deduced  from  the  tran- 
sit of  any  star  whose  right  ascension  is  known ;  but  the  places 
of  all  stars  contained  in  the  catalogues  are  not  equally  well  de- 
termined ;  and  it  is  obviously  proper  that  the  stars  whose  places 
are  best  determined  should  be  preferred  for  this  purpose.  The 
places  of  the  100  stars  in  the  Nautical  Almanac  are  considered 
to  be  better  known  than  any  others.  At  the  Greenwich  Obser- 
vatory, the  error  of  the  clock  is  determined  exclusively  by  the 
Nautical  Almanac  stars ;  and  only  those  are  used  whose  decli- 
nation is  less  than  40  degrees. 

At  the  Oxford  Observatory,  the  stars  used  for  finding  the  clock 
error  are  chiefly  the  Nautical  Almanac  stars,  but  occasionally 
other  stars  are  employed. 

At  the  Edinburgh  Observatory,  only  Nautical  Almanac  stars 
are  used  for  determining  the  correction  of  the  clock,  and  of  this 
list  only  those  are  employed  whose  places  are  considered  to  be 
best  determined, 


CHAPTER  VI. 

LATITUDE. 

(172.)  THE  latitude  of  a  place  is  equal  to  the  elevation  of  the 
pole  above  the  horizon,  and  this  altitude  could  be  easily  determ- 
ined if  the  pole  were  a  visible  point.  But  as  there  is  no  star 
exactly  at  the  pole,  its  position  must  be  determined  by  observa- 
tions of  stars  at  a  distance  from  it. 

FIRST    METHOD. 

By  transits  of  a  circumpolar  star  both  above  and  below 
the  pole. 

The  best  method  of  determining  the  latitude  of  a  place,  so  as 
to  be  independent  of  the  declination  of  the  star  observed,  and 
also  as  free  as  possible  from  the  errors  of  refraction,  is  by  obser- 
vations of  a  circumpolar  star  at  the  time  of  its  upper  and  lower 
culminations.  These  observations  may  be  made  by  means  of 
a  mural  circle,  or  any  graduated 
circle. 

Let  HZPO  represent  a  meridian, 
HO  the  horizon  of  the  place  of  ob- 
servation, P  the  place  of  the  pole, 
and  ABCD  the  circle  described  by 
a  circumpolar  star  in  its  diurnal  motion.  The  elevation  of  the 
pole  PO  is  equal  to  half  the  sum  of  AO  and  CO,  corrected  for 
refraction. 

Let  A  and  Ax  represent  the  altitudes  of  a  circumpolar  star  at 
its  upper  and  lower  culminations  ;  also,  let  r  and  rf  be  the  re- 
fractions corresponding  to  these  altitudes  ;  then 


both  altitudes  being  measured  from  the  north  horizon. 

If  zenith  distances  instead  of  altitudes  are  observed,  the  co- 
latitude  will  be, 


The  refraction  is  derived  from  Table  VIII. 


138  PRACTICAL    ASTRONOMY. 

Examples.  The  following  observations  were  made  at  Green- 
wich Observatory : 
Polaris,  May  9,  1851. 

Observed  Altitude.  Refraction.  True  Altitude. 

Lower  culmination,  50°    Ox   8XX.49-  48XX.08=   49°  59X  20XX.41 
Upper  culmination,  52°  58X38XX.31-   42XX.16=   52°57X56XX.15 

Sum        =  102°57X16XX.56 
Latitude^  51°  28X 38XX.28 
<5  Ursse  Minoris,  January  22,  1851. 

Lower  culmination,  48°    5X18XX.60-   53/x.80=  48°    4X24XX.80 
Upper  culmination,  54°  53X  33XX.22  -  42XX.49=  54°  52X  50XX.73 

Sum        =  102°57X15XX.53 
Latitude^   51°  28X37XX.77 
ft  Cephei,  March  17,  1847. 

Low.  culmination,  31°  23X32XX.85-1X35XX.22=:  31°21X57XX.63 
Upp.  culmination,   71°  35X  36XX.53  -  Ox  19/x.22  =  _71°_35^17^.31 

Latitude  =   51°  28X  37/x.47 
a  Cephei,  March  17,  1847. 

Low.  culmination,  23°  27X    6XX.03-2X13XX.53=:  23°  24X  52XX.50 
Upp.  culmination,   79°  32X  33XX.71  -  Ox  10XX.66  =  79°  32X  23XX.Q5 

Sum        =  102°57X15XX.55 
Latitude^   51°  28X 37XX.77 
Capella,  June  14,  1837. 

Low.  culmination,     7°  25'   0/X.72-6X52/X.78^     7°  18X   7XX.94 
Upp.  culmination,   95°  39X   2XX.42+       5XX.49=   95°  39X   7XX.91 

Sum        =102°57X15/X.85 
Latitude^   51°  28X  37XX.92 

As  the  refraction  for  the  last  star  is  large,  the  result  of  that 
observation  is  less  reliable  than  the  others.  Those  stars  are  ac- 
cordingly to  be  preferred  whose  polar  distance  is  the  least. 

SECOND    METHOD. 

(173.)  By  simple  meridian  alti- 
tudes. 

Let  PZH  represent  the  meridian 
of  the  place  of  observation,  HO  the 
horizon,  Z  the  zenith,  P  the  place  of 
°    the  pole,  EQ,  the  equator,  S  or  Sx  a 


LATITUDE.  139 

star  on  the  meridian,  SE  or  S'E  its  declination  (<5),  SP  or  S'P  its 
distance  from  the  pole  (d),  which  is  the  complement  of  6  ;  the 
arc  EH  is  the  complement  of  the  latitude  (</>),  or  90°  —  0. 

We  measure  the  altitude  (A)  of  the  object  S  or  Sx,  or  its  zenith 
distance  (Z),  and  correct  it  for  refraction  and  parallax,  if  the 
parallax  is  appreciable.  Then  it  is  evident  that 


These  two  equations  are  included  in  the  same  expression  by 
regarding  the  declination  negative  when  it  is  south  of  the  equa- 
tor. Thus, 

90°-0=A-d, 
or  <j)  =  9Q°  +  6-A.. 

But  Z  =  90°-A. 

Hence  0  =  <f  +  Z, 

for  stars  which  culminate  south  of  the  zenith,  where  6  must  have 
the  negative  sign  when  the  declination  is  south. 

If  the  star  passes  the  meridian  between  the  north  pole  and 
the  zenith,  as,  for  example,  at  B,  then  we  shall  have 

PO^BO-BP; 
that  is, 

</>  =  A—  d. 

But  A=90°-Z,  and  d  =90°  -6. 

Hence  (f>  —  6—  Z. 

If  the  star  passes  the  meridian  helow  the  north  pole,  then  we 
shall  have 

PO  =  CO-fPC; 
that  is,  0=A+d=180°-<i-Z. 

Hence  we  shall  have 

0  =  d-hZ  if  the  observations  be  made  to  the  south  ; 

$=S—  Z  if  to  the  north,  above  the  pole  ; 

0  =  180°-  (<S+Z)  if  to  the  north,  below  the  pole. 

The  following  observations  were  made  at  Greenwich  in  1851  : 

Stars  South  of  the  Zenith. 
February  10,  1851. 

Observed  Zenith  Distance.  Refraction.  True  Zenith  Distance. 

Pollux  .....  23°    5'24".05+     25".  80=     23°    5X  49".85 

Star's  declination^     28°  22/  47".7Q 
Latitude  ==     51°  28/  37/x.55 


140  PRACTICAL   ASTRONOMY. 

July  10,  1851. 

Observed  Zenith  Distance.      Refraction.  True  Zenith  Distance. 

Antares  ....  77°  30'  ll//.54+4/  15/x.48^     77°  34'  27/x.02 

Star's  declination  =  -26°    5X  48X/.5Q 
Latitude  =     51°  28'  38x/.52 

June  30,  1851. 

Sun.  Observed  zenith  distance  of  upper  limb,      27°  59'  39/x.53 

semi-diameter,    +15/46//.05 

refraction,  -f297/.49 

parallax,  -3/7.93 

true  zenith  distance  of  centre  =     28°  15X  51/7.14 

Sun's  declination  =  +23°  12X  47/x.3Q 

Latitude  =     51°  287  38/7.44 

The  method  of  computing  the  sun's  parallax  will  be  explained 
in  Article  206. 

Star  North  of  the  Zenith,  and  above  the  Pole. 
June  29,  1851. 

Observed  Zenith  Distance.    Refraction.  True  Zenith  Distance. 

a  TJrsse  Majoris,      11°    4X  39".01-r-10//96=     11°    4' 49."97 

Star's  declination  =  +62°  33X  26/x.69 
Latitude  =     51°  28X  36/x.72 

Star  below  the  Pole. 
January  27, 1851. 

Observed  Zenith  Distance.        Refraction.  True  Zenith  Distance. 

j3  Ursa*  Minoris,  53°  447  24//.60  +  l"  20/x.69=   53°  45X  45/x.29 

Star's  declination  =  74°  45X  37x/.79 
Sum  =128°  31' 23^.08 
Latitude  =  51°  28X  36/x.92 

THIRD    METHOD. 

(174.)  By  circum-meridian  altitudes. 

The  preceding  method  gives  but  one  value  of  the  latitude,  be- 
cause the  star  can  only  be  observed  at  the  instant  when  it  crosses 
the  meridian.  But  where  the  observer  is  furnished  with  an  al- 
titude and  azimuth  instrument,  a  repeating  circle  or  sextant,  we 
may  render  any  number  of  observations  made  on  each  side  of 
the  meridian,  and  at  a  short  distance  from  it,  equal  in  accuracy 
to  those  which  are  made  at  the  moment  of  culmination.  For 


LATITUDE.  141 

this  purpose,  we  must  know  the  distance  (in  time)  of  the  star 
from  the  meridian  at  the  instant  of  each  observation,  and  we 
can  compute  the  correction  which  ought  to  be  applied  to  the 
zenith  distance  observed. 

Let  P  be  the  pole,  Z  the  zenith 
of  the  place  of  observation,  PZM  a 
meridian,  S  a  star  near  to  the  me- 
ridian, M  the  point  where  this  star 
crosses  the  meridian,  and  PS  an 
hour  circle  passing  through  the  star. 
Suppose  the  zenith  distance,  ZS,  of  the  star  has  been  measured, 
and  corrected  for  refraction,  and  also  for  parallax,  when  the  sun 
or  a  planet  has  been  observed  ;  it  is  required  to  compute  the 
zenith  distance,  ZM,  of  the  star  when  on  the  meridian,  Now 
from  the  figure  we  perceive  that 

PS  =  90°-  d 


the  zenith  distance  of  the  star  on  the  meridian. 

With  Z  as  a  centre,  describe  the  arc  SN,  and  the  point  N  will 
be  at  the  same  altitude  as  S.  It  is  required  to  compute  MN=x, 
the  quantity  which  the  star  must  rise  from  S,  before  it  reaches 
the  meridian. 

By  Trig.,  Art.  225, 

cos.  &  =  cos.  b  cos.  c  +  sin.  b  sin.  c  cos.  A. 

But  cos.  A  =1  —  2  sin.  2i  A.     Trig.,  Art.  74. 

Hence 

cos.  &  =  cos.  b  cos.  c-f-sin.  b  sin.  c  —  2  sin.  b  sin.  c  sin.  2JA. 

Also,  cos.  b  cos.  c  +  sin.  b  sin.  c=cos.  (b  —  c).     Trig.,  Art.  72. 

Hence  cos.  a  =  cos.  (b  —  c)  —  2  sin.  b  sin.  c  sin.  2£A. 

Applying  this  formula  to  the  triangle  PZS,  and  representing 
the  angle  ZPS  by  P,  we  have 

cos.  ZS  =  cos.  (PS-PZ)-2  sin.  PZ  sin.  PS  sin.  2JP 

=  cos.  Z  —  2  cos.  0  cos.  6  sin.  2JP  ......  (1) 

But  ZS  =  ZM-hz. 

Hence 

cos.  ZS^cos.  ZM  cos.  a;—  sin.  ZM  sin.  x.     Trig.,  Art.  72. 

But,  since  x  is  supposed  to  be  a  small  arc,  we  may  put 


142  PRACTICAL   ASTRONOMY. 

x2 
and  cos.  x=l— — +,  etc.     Calculus,  p.  174. 

Hence  we  obtain 

cos.  ZS^cos.  ZM(1-|-  +  ,  etc.) -a  sin.  ZM 

=  cos.  Z  —  %x2  cos.  Z—x  sin.  Z. 
Therefore  equation  (1)  becomes 

cos.  Z  —  jrc2  cos.  Z— a;  sin.  Z=cos.  Z— 2  cos.  0  cos.  tf  sin.  2^P, 
or  %x2  cos.  Z+a;  sin.  Z  =  2  cos.  0  cos.  (5  sin.  2iP  .  .  (2) 

If  we  neglect  the  term  containing  a;2,  and  suppose  x  to  be  ex- 
pressed in  seconds,  we  shall  hive 

_2  sin.  2jP  cos,  ft  cos.  6 

sin.  1"  sin.  Z 

which  formula  is  sufficiently  accurate,  when  the  hour  angle  does 
not  exceed  ten  minutes.  If  a  further  approximation  is  required, 
it  may  be  obtained  as  follows : 

Divide  equation  (2)  by  sin.  Z,  and  we  obtain 

,    „     2  sin.  2JP  cos.  d)  cos.  6 

•  cot.  Z  = —. — =-^ . 

sin.  L 

Represent  the  second  member  of  this  equation  by  B,  and 
£  cot.  Z  by  A,  then  x  -f-  Aa;2  =  B, 

or  x  =  ~B-Ax2. 

But  B  is  the  approximate  value  of  x  before  found ;  hence,  for 
a  second  approximation,  we  shall  have 

or,  supposing  x  to  be  expressed  in  seconds, 
_2  sin.  2^P  cos.  0  cos.  6     /sin.  2JP  cos.  </>  cos.  <$Y  2  cot.  Z      ,<,. 

sin.  lx/  sin.  Z  \  sin.  Z  /    sin.  V 

which  is  the  correction  to  be  subtracted  from  the  zenith  distan- 
ces observed  near  the  meridian,  for  an  upper  culmination,  in  or- 
der to  obtain  the  true  meridional  zenith  distance. 

Ex.  1.  On  the  23d  of  February,  1850,  the  zenith  distance  of 
a  Orionis  was  observed  at  Greenwich,  20m.  26.25s.  before  com- 
ing to  the  meridian,  to  be  44°  18X  30XX.31,  the  declination  of  the 
star  being  7°  22X  14x/.74  N.  Required  the  reduction  to  the  me- 
ridian and  the  resulting  latitude. 

Here  P=20m.  26.25s.;  therefore  iP  =  10m.  13.12s.,  which, 
reduced  to  arc,  is  2°  33X  16x/.9. 


LATITUDE.  143 

sin.        2°  33X  16". 9  =  8.649071 

8.649071  m2= 4.4926 

cos.  </>,  51°  28'  38"    =9.794366  cot.  z= 0.0135 

cos.  d,    7°  22"  15XX    =9.996396  2  cosec.  l/x  =  5.6155 

cosec.*,  44°    6X23/X    =0.157394  lxx.32=0.1216 

w  =  7.246298 
2  cosec.  1"      =  5.615453 
727".37  =  2.861753 
Therefore 

x = 727XX.37  - l/x.32  =  726XX.05. 
Hence  we  have 

Observed  zenith  distance  =44°  18X  30XX.31 
Reduction  to  the  meridian  =  -12K_  6/x.05 
Corrected  zenith  distance  =44°  W~24S7.26 
Star's  declination  =  7°  22X  14/x.74 

Latitude  =51°  28'  SlFjOO 

(175.)  To  diminish  the  labor  of  these  reductions,  Table  X.  has 

2  sin  2AP 

been  computed,  in  which  Part  I.  gives  the  value  of '—2- — ; 

sin.  l/x 

and  the  argument  of  the  table  is  the  distance  (in  time)  of  the 
star  from  the  meridian.  This  value  (or  the  sum  of  those  values 
divided  by  the  number  of  observations,  if  more  than  one  observa- 
tion has  been  made)  must  be  multiplied  by  — ~- ' — .  Part 

sin.  Z 

2  sin  4J*P 

II.  of  the  table  contains  the  value  of  — . — V2— >  which  must  be 

sin.  l/x 

multiplied  by  I  -     .  I  cot.  Z.     This  second  correction  may 

generally  be  omitted  when  the  distance  from  the  meridian  does 
not  exceed  ten  minutes. 

Ex.  2.  The  following  observations  of  Polaris  at  its  upper  cul- 
mination were  made  at  Washington  Observatory,  November  25, 
1845,  the  altitude  of  the  star  having  been  observed  at  each  vert- 
ical wire  of  the  mural  circle.  In  the  following  table,  column 
first  shows  the  wire  at  which  the  observation  was  made,  column 
second  shows  the  hour  angle  of  the  star  from  the  meridian,  and 
column  third  shows  the  observed  zenith  distances  corrected  for 
error  of  runs : 


144 


PRACTICAL   ASTRONOMY. 


Wire. 

Hour  angles. 

Zenith  Distances. 

Table  X. 

m.         s. 

0               /                      " 

" 

1 

29  25.5 

49  35  27.92 

1697.65 

2 

19  34.5 

35    2.86 

751.95 

3 

9  49.5 

34  48.12 

189.50 

4 

0  13.5 

34  43.74 

0.10 

5 

9  33.5 

34  48.27 

179.35 

6 

19  13.5 

35     2.56 

725.25 

7 

29     0.5 

35  28.07 

1649.95 

Means 

49  35     3.07 

741.96 

Column  fourth  contains  the  numbers  from  Table  X.  corre- 
sponding to  the  hour  angles  in  column  second.  The  mean  of 
these  numbers  is  741".96.,  The  reduction  to  the  meridian  is 
then  computed  as  follows : 

741//.96  =  2.87038 

cos.  </>,  38°  53'  39"  =  9.89115 

cos.  d,  88°  -m  34"  =  8.42000 

cosec.  *,  49°  35'  55/x  =  0.11832 

Reduction  =  19/x.95  =  1.29985 


The  latitude  will  then  be  obtained  as  follows : 
Mean  of  observed  zenith  distances =49°  35' 


3".07 


Reduction  to  the  meridian 

Refraction 

Corrected  zenith  distance 

Star's  decimation 

Latitude 


-19/x.95 
V  llx/.08 


=  49°  35/  54X/.20 
=  88°  29/  34X/.15 


=  38°  53'  39x/.95 
(176.)  In  the  preceding  reductions  it  is  necessary  to  know  the 
distance  (in  time)  of  the  sun  or  star  from  the  meridian,  or  the 
hour  angles,  at  the  moment  of  each  observation.  These  hour 
angles  are  determined  by  the  chronometer ;  and  it  is  desirable 
that  its  motion  should  correspond  to  that  of  the  object  observed ; 
that  is,  if  the  sun  be  the  object,  the  chronometer  should  be  ad- 
justed to  mean  solar  time  ;  and  if  a  star  be  the  object,  the  chro- 
nometer should  be  adjusted  to  sidereal  time.  This,  however,  is 
not  necessary,  since  a  correction  may  be  readily  applied  so  as  to 
reduce  the  rate  either  from  mean  solar  to  sidereal,  or  from  si- 
dereal to  mean  solar  time.  A  further  correction  is  also  necessary 
in  all  cases  where  the  chronometer  has  a  gaining  or  a  losing  rate 
on  either  mean  solar  or  sidereal  time.  This  correction  is  ob- 
tained in  the  following  manner : 


LATITUDE.  145 

If  the  clock  in  24  hours  loses  r  seconds,  then,  instead  of  86400 
beats  in  a  day,  it  will  make  only  86400—  r.     The  true  value  of 
an  hour  angle,  P,  noted  by  such  a  clock,  is 
86400  /  r 


that  is,  the  observed  hour  angle  should  be  multiplied  by  the 
factor 

1+  86400  -r 

The  principal  term  of  formula  (3)  for  the  reduction  on  page 
142  contains  the  factor  sin.  2£P,  which  must  therefore  be  mul- 
tiplied by  the  square  of  the  above  factor,  which  is  nearly  equal  to 

1+       2r 
^  86400  -r 

If  the  clock  indicates  mean  solar  time,  and  we  are  observing  a 
star,  the  clock  loses  235.909s.  in  24  hours,  when  compared  with 
the  progress  of  the  star,  and  we  must  take  r—  235.909s.,  and  the 
preceding  factor  becomes 

1.005476, 
and  its  logarithm  is 

0.0023715. 

In  a  similar  manner  we  obtain  the  correction  for  a  loss  or  gain 
of  1,  2,  3,  etc.,  seconds  per  day  of  the  chronometer.  This  cor- 
rection has  been  computed,  and  is  appended  to  Table  X.,  which 
gives  the  logarithm  of  the  factor  to  be  employed  for  a  daily  rate 
of  the  clock  or  chronometer,  amounting  to  ±  30  seconds.  These 
values  are  in  all  cases  additive. 

Ex.  3.  On  the  18th  of  October,  1841,  near  the  River  St.  John's, 
in  latitude  about  46°  53X  30/x,  the  following  observations  were 
made  on  the  star  a  Ceti,  declination  3°  28X  8x/.2  N.  Column 
first  shows  the  number  of  the  observation,  column  second  shows 
the  hour  angle  of  the  star  from  the  meridian,  and  column  third 
shows  the  observed  altitude  corrected  for  the  error  of  the  sex- 
tant. The  chronometer  was  regulated  to  mean  solar  time,  and 
had  a  daily  losing  rate  of  2.7s. 

K 


146 


PRACTICAL   ASTRONOMY. 


.      : 

Table 

X. 

Obs. 

Hour  Angles. 

Part  I. 

Part  II. 

m.        s. 

o           /           // 

1 

14  30.0 

46  28  45 

412.70 

0.41 

2 

10  50.2 

46  31  45 

230.54 

0.13 

3 

5     0.0 

46  34  35 

49.10 

0.01 

4 

1  59.7 

46  35  25 

7.77 

0.00 

5 

2     3.0 

46  35  30 

8.20 

0.00 

6 

5     6.5 

46  34  45 

51.25 

0.01 

7 

7  21.7 

46  33  50 

106.45 

0.03 

8 

13     7.3 

46  30     0 

337.97 

0.28 

9 

16  40.1 

46  26  35 

545.39 

0.72 

Means 

>  . 

46  32  21.1 

194.37 

0.18 

Column  fourth  contains  the  numbers  from  Part  I.,  Table  X., 
corresponding  to  the  hour  angles  in  column  second.     Column 
fifth  contains  the  numbers  from  Part  II.,  Table  X.     The  reduc- 
tion to  the  meridian  is  then  computed  as  follows : 
cos.  0,  46°  53'  30"= 9.834662 
cos.  d,    3°  28'    8"  =  9.999204 
cosec.  z9  43°  25'  22"  =  0.162805 


m2  =  9.993 
cot.  *  =  0.023 
0.18  =  9.255 
0.19  =  9.271 


m  =  9.996671 

194/x.37= 2.288629 

On  account  of  mean  solar  time  =  0.002371 

On  account  of  rate  of  clock       =0.000027 

193/x.95  =  2.287698 

Therefore        x = 193/x.95  -  Ox/.19  =  193".76. 
Hence  we  have  the  following  results : 

Observed  zenith  distance  =43°  27X  38/x.9 
Reduction  to  the  meridian  =  —  3X  13x/.8 
Refraction  +  55/x.8 

Corrected  zenith  distance  =43°  25X  20/x.9 
Star's  declination  =  3°  287  8/x.2 

Latitude  =46°  53'  29/x.l 

(177.)  When  the  sun  is  the  object  observed,  we  must  take 
into  account  the  change  of  declination  during  the  interval  of  the 
observations ;  for  the  observed  altitude,  corrected  in  the  manner 
before  explained,  will  not  be  equal  to  the  meridian  altitude,  but 
will  differ  from  it  by  the  change  in  the  sun's  declination.  Let 
the  change  of  the  sun's  declination  in  one  minute  of  time  be  de- 
noted by  dd,  which  is  positive  when  the  sun  is  approaching  the 


L  ATITUDE. 


147 


elevated  pole ;  and  if  P  is  the  sun's  hour  angle  at  the  time  of 
observation,  which  is  negative  before  the  sun  arrives  at  the  me- 
ridian and  afterward  positive,  the  whole  change  of  declination  is 
P<^<5,  which  is  the  correction  to  be  applied  to  the  altitude  found 
by  Art.  176  to  obtain  the  true  meridian  altitude.  When  sev- 
eral observations  have  been  made,  the  mean  of  the  values  found 
by  Art.  176  is  to  be  diminished  by  the  mean  of  the  values  of 
Pdd.  But  the  hour  angles  have  contrary  signs  on  opposite  sides 
of  the  meridian;  hence,  if  we  make  E=the  sum  of  the  hour 
angles  observed  on  the  east  side  of  the  meridian,  and  W^the 
sum  of  the  hour  angles  observed  on  the  west  side,  (E—  W)d6 
will  be  the  correction  for  the  sum  of  the  observed  distances.  If 
we  make  n = the  number  of  the  observations,  the  mean  correction 
to  be  applied  to  the  mean  of  all  the  observed  zenith  distances 
will  be 

^(B-W), 

VI 

where  E  and  W  are  expressed  in  minutes  of  time. 

Ex.  4.  At  a  station  in  Lat.  51°  32'  N.  nearly,  the  correct 
central  altitudes  of  the  sun  on  the  llth  of  March  were  determ- 
ined by  observation,  as  follows : 


Altitudes. 

Hour  Angles. 

By  Table  X. 

°         '         " 

m.     s. 

" 

34  54  46 

9  41  E. 

184.1 

55  26 

8  19  E. 

135.8 

56     8 

6  39  E. 

86.8 

56  31 

5  16  E. 

54.5 

56  53 

3  49  E. 

28.6 

57     6 

2  47  E. 

15.2 

57  18 

0  19  W. 

0.2 

57  11 

2     5  W. 

8.5 

57     3 

3     9  W. 

19.5 

56  48 

4  36  W. 

41.5 

56  26 

6     8  W. 

73.9 

The  sun's  meridian  declination  was  3°  30'  38"  S.,  and  it  was 
decreasing  at  the  rate  of  Ox/.98  in  a  minute.  "What  was  the  true 
latitude  ? 

Entering  Table  X.  with  the  hour  angles  given  above,  we  ob- 
tain the  values  set  down  in  the  last  column,  the  sum  of  which, 
being  divided  by  11,  will  give  58x/.96  ;  whence,  by  formula  (3), 
page  142,  we  obtain  the  reduction  to  the  meridian,  44".7. 


148  PRACTICAL   ASTRONOMY. 

The  sum  of  the  eastern  hour  angles,  diminished  by  the  sum 
of  the  western,  and  divided  by  11,  gives  1m.  50.3s.,  which,  mul- 
tiplied by  0/x.98,  gives  l/x.8  for  the  correction  for  change  of  dec- 
lination. 

Hence  we  have  the  following  results : 

Mean  of  the  observed  altitudes         =     34°  56'  30". 5 
Reduction  to  the  meridian  -f  44x/.7 

Correction  for  change  of  declination  =  +  1".8 

Concluded  meridian  altitude  =     34°  57'  17/x.O 

Zenith  distance  =     55°    2'  43" 

Sun's  declination  =  -   3°  30X  38" 


Latitude  =     51°  32X    5/x  N. 

FOURTH    METHOD. 

(178.)  By  a  single  altitud&Jtfie  time  of  observation  being 
known.  '* 

Liet  Z  be  the  zenith  of  the  ob- 
server, P  the  pole,  S  a  star  whose 
altitude  is  measured  at  a  known 
instant   of   time.      Then,   in   the 
spherical   triangle  ZPS,  we   have 
~J°    givenPS=90°-d,ZS=:Z,andthe 
hour  angle  ZPS,  to  find  PZ. 

From  S  let  fall  the  perpendicular  SM  upon  PZ  produced. 
Then,  by  Napier's  rule,  we  shall  have 

R.  cos.  P=tang.  PM  cot.  PS  =  tang.  PM  tang.  d. 
Hence  tang.  PM—  cos.  P  cot.  6  ........  (1) 


Also,  Trig.,  Art.  216, 

cos.  PM  :  cos.  ZM  ::  cos.  PS  :  cos.  ZS, 
or  cos.  PM  :  sin.  (PM  +  </>)  :  :  sin.  6  :  cos.  Z. 

TT  •       /r>T\/r   ,      \       cos-  2  COS.  PM  . 

Hence  sin.  (PM.-\-d>)=—   —.  -  -    .....  (2) 

sin.  o 

Equation  (1)  furnishes  the  value  of  PM,  and  equation  (2) 
furnishes  the  value  of  PM  +  </>.  The  difference  between  these 
quantities  is  </>,  the  latitude  required. 

Ex.  1.  At  Ih.  14m.  11.6s.  apparent  time,  the  true  altitude 
of  the  sun  was  33°  40X  357/.5,  and  his  declination  5°  15X  28X/.0  S. 
Required  the  latitude  of  the  place. 


LATITUDE.  149 

By  equation  (1),  cos.  18°  32X  54X/          =  9.976834 

cot.    5°  15X  28XX          =1.0360997* 
PM=95°  32X  39X/  tang.  =  1.012933ra 
By  equation  (2), 

cos.  56°  19X  24XX.5  =  9.743904 
cos.  PM  =  8.985035ra 
cosec.  J=  1.037930ft 
PM  +  0  =  144°  13X  28XX  sin.  =  9.766869 
=   95°  32X  39XX 


0=  48°  40X  49XX 

.  2.  At  a  place  in  Lat.  42°  34X  N.  nearly,  the  altitude  of 
Aldebaran  (Dec.  16°  12X  26/x  N.)  was  found  by  observation  to 
be  39°  2X  10/x,  when  its  hour  angle  was  3h.  25m.  40s.  What 
was  the  latitude  of  the  place  ?  Ans.  42°  34X  56". 

Ex.  3.  At  a  place  in  Lat.  41°  25X  nearly,  the  altitude  of  Reg- 
ulus  (Dec.  12°  41X  18/x  N.)  was  found  by  observation  to  be  41° 
5X  20/x,  when  its  hour  angle  was  3h.  2m.  21s.  "What  was  the 
latitude  of  the  place  ?  Ans.  41°  25X  47XX. 

Ex.  4.  On  the  27th  of  February,  1850,  the  zenith  distance  of 
Procyon  (Dec.  5°  36X  6/x.7  N.)  was  observed  at  Greenwich  to  be 
48°  48X  34/x.06,  when  its  hour  angle  from  the  meridian  was  Ih. 
20m.  18.13s.  It  is  required  to  deduce  the  latitude  from  this  ob- 
servation. Ans. 

This  method  is  deficient  in  accuracy  when  the  observations 
are  made  far  from  the  meridian,  because  a  small  error  in  the 
hour  angle  produces  a  large  error  in  the  computed  value  of  the 
latitude.  The  observations  should,  therefore,  al- 
ways  be  made  as  near  as  possible  to  the  meridian. 

FIFTH    METHOD. 

(179.)  By  observations  of  the  pole  star  at  any 
time  of  the  day. 

Let  P  be  the  pole,  Z  the  zenith,  ZPN  the  merid- 
ian of  the  place  of  observation,  and  S  the  pole  star 
in  any  point  of  its  diurnal  circle,  SBSr.  Then  we 
shall  have  ZP  =  90°  -  0,  ZS  =  90°  -  H,  H  being  the 
observed  height  of  the  star  corrected  for  refraction. 
Represent  the  polar  distance,  PS,  by  d.  Since 
the  arc  d  is  at  present  less  than  90X,  the  sides  ZP, 


150 


PRACTICAL   ASTRONOMY. 


ZS,  differ  only  by  a  small  arc,  x,  which  we  propose 
to  calculate. 

Let     ZP-ZS=o;;  that  is,  H-<j>=x, 

Represent  the  hour  angle  ZPS  by  P.     The  spher- 
ical triangle  ZPS  furnishes  by  Trig.,  Art.  225, 

cos.  ZS=cos.  PS  cos.  PZ+sin.  PS  sin.  PZ  cos.  P, 
or 

sin.  H  =  cos.  d  sin.  (H — x)  +  sin.  d  cos.  (H — x)  cos.  P. 
By  substituting  the  values  of  sin.  (H — x),  and 
cos.  (H— z),  Trig.,  Art.  72,  and  dividing  the  whole 
equation  by  sin.  H,  we  obtain 
l=cos.  d  (cos.  a;  — sin.  x  cot.  H) 

+  sin.  d  (cos.  x  cot.  H  +  sin.  x)  cos.  P, 
l=cos.  x  (cos.  d -\-sin..  d  cot.  H  cos.  P) 

—sin.  x  (cos.  d  cot.  H  — sin.  d  cos.  P). 
Let  us  put 

a  =  cos.  d -\-siu.  d  cot.  H  cos.  P, 
b— cos.  d  cot.  H— sin.  d  cos.  P, 
and  we  have 

\  —  a  cos.  x— b  sin.  x (1) 

But  by  Calculus,  Art.  228, 

sin.  #=#-|_+Jr--,etc. 


Therefore 


cos.  y  =  l- +-,  etc. 


d2     d3 

a  —  \-\rd  cos.  P  cot.  H—  —  —  —  -  cos.  P  cot.  H+,  etc. 
2      o 


b  =  cot.  H-d  cos.  P—      cot.  H+r  cos.  P  +  ,  etc. 
Let  us  now  assume 


(2) 


where  A,  B,  and  C  represent  unknown  coefficients  independent 
ofd. 


Then  we  shall  have 
cos.  x  =  1 


,  etc. 


LATITUDE.  151 


sin.  x=Ad+Bd2+c-~    P  +  ,  etc. 

Substituting  in  equation  (1)  the  values  of  &,  £,  sin.  #,  and 
cos.  x,  arranging  the  terms  in  the  order  of  the  powers  of  d,  and 
retaining  all  the  terms  which  contain  the  first  three  powers  of  d, 
we  obtain 

1  =  1  +  cos.  p  cot.  H.^-—  -  -  cos.  P  cot.  H 


--  -  ---  —  cos.  P  cot.  H—  ABd? 

*/£         £ 

A/73 
-A  cot.  H.d+A  cos.  P.d*  +  -  -  cot.  H+Bd3  cos.  P 


-B  cot.  H.6Z2-fc-^cot.  H.d3. 


Since  this  equation  must  be  verified  by  any  value  of  d,  the 
terms  involving  the  same  powers  of  d  must  cancel  each  other. 
Algebra,  Art.  300. 

Hence, 

First,     cos.  P  cot.  H— A  cot.  H=0 ;  whence  A=cos.  P. 


Second.        —  £• — ^-+A  cos.  P— B  cot.  H=0. 

Therefore, 

cos.  2P     .     cos.  2P  —  1         sin.  2P 
B  cot.  H=cos.  2P p i  = = — . 

sin.  2P  ,         -u- 
Hence  B  = —  tang.  H. 

cos.  P     A2  cos.  P  ,  A     /       A3\ 
Third. g __+__^C_-J=0. 

"Whence,  substituting  the  value  of  A  already  found, 

3C:=COS.  P-COS.  3P 

=cos.  P(l-cos.  2P) 
=cos.  P  sin.  2P. 
Therefore, 

C=.j  cos.  P  sin.  2P. 
Substituting  these  values  in  equation  (2),  we  obtain 

x  —  d  cos.  P  — i  sin.  2P  tang.  H.sP+i  cos.  P  sin.  2PoP; 
or,  multiplying  by  sin.  l/x,  in  order  that  x  may  be  expressed  in 
seconds  of  arc,  we  have 


152  PRACTICAL   ASTRONOMY. 

<j>=tt-d  cos.  P+i  sin.  1"  (d  sin.  P)2  tang.  H 

-i  sin.  21/X  (d  cos.  P)  (d  sin.  P)2  ......  (3) 

The  last  term  of  this  equation  never  amounts  to  half  a  sec- 
ond, and  may  therefore  generally  be  omitted. 

Ex.  1.  The  altitude  of  the  pole  star  being  found  46°  IT  28X/, 
the  hour  angle  5h.  42m.  4.4s.  from  the  upper  culmination,  and 
the  polar  distance  1°  28'  7/x.68  ;  required  the  latitude  of  the 
place. 

Computation  by  formula  (3),  v 

d=  5287x/.68  =  3.72327          d=  3.7233       d  cos.  P  =  2.616 

cos.  P,  85°  31'G"  =  8.89287   sin.  P  =  9.9987    (d  sin.  P)2  =  7.444 

413//.2=  2.61614  3.7220     £  sin.  2lx/=  8.894 

3.7220  Ox/.l=:a954 

tang.  H,  0.0196 
j  sin.  1"  =  4.3845 
70".5  =  1.8481 

.^  Result. 

Observed  altitude,  H=:46°  17X  28/x.O 

first  correction,        =       —  6X  53/x.2 

second  correction,    —       -f  V  10/x.5 

third  correction,       =  —  Ox/.l 

Latitude  =46°  11'  45/x.2 

The  computation  may  also  be  performed  by  the  formulas  of 
Art.  178. 

cos.  P=  8.892874 
cot.  6=  8.408935 
=  6/53//.3  tang.  =  7.301809 

sin.  H=r  9.8590542 
cos.  PM  =  9.9999991 
cosec.  d= 


=46°  187  38/x.5=  9.8591960 


Latitude  =46°  11'  45x/.2 

The  method  of  Art.  178  is  about  as  convenient  as  the  one 
here  explained,  except  that,  when  great  accuracy  is  demanded, 
the  former  method  requires  logarithms  to  seven  places. 

Ex.  2.  The  altitude  of  the  pole  star  being  43°  2X  38/x  when 


LATITUDE.  153 

the  hour  angle  was  76°  0'  2"  from  the  upper  culmination,  and 
its  Dec.  88°  31/  52/x.32;  required  the  latitude  of  the  observer. 

Ans.  42°  42'  18".  2. 

Ex.  3.  The  altitude  of  the  pole  star  being  found  39°  V  39", 
the  hour  angle  5h.  36m.  41s.  from  the  upper  culmination,  and 
the  polar  distance  1°  28'  7x/.68  ;  required  the  latitude  of  the 
place.  Ans.  38°  53'  36".2. 

SIXTH    METHOD. 

(180.)  By  observing  the  difference  of  the  meridional  zenith 
distances  of  two  stars  on  opposite  sides  of  the  zenith. 

If  we  select  two  stars  whose  places  are  well  known,  one  of 
which  culminates  to  the  north,  and  the  other  to  the  south  of 
the  observer,  at  nearly  the  same  distances  from  the  zenith,  and 
within  a  short  interval  of  time,  and  measure  accurately  the 
difference  of  their  zenith  distances,  the  latitude  of  the  place  of 
observation  may  thence  be  easily  deduced.  If  we  represent  the 
zenith  distance  of  the  northern  star  by  Zn,  and  that  of  the  south- 
ern star  by  Zg  ;  also  the  declination  of  the  northern  star  by  <?n, 
and  that  of  the  southern  star  by  <58,  then,  by  Art.  173,  we  shall 
have 


Hence  20=<?B+dn  +  Z,—  Zn; 

that  is,  the  sum  of  the  declinations  of  the  two  stars  (which  are 
given  by  the  catalogue),  added  to  the  difference  of  their  zenith 
distances,  gives  twice  the  latitude  of  the  place. 

(181.)  The  instrument  employed  in  measuring  the  difference 
of  the  zenith  distances  is  called  the  Zenith  Telescope.  The 
figure  on  the  next  page  represents  this  instrument  in  the  form 
now  used  in  the  coast  survey  of  the  United  States. 

A,  A  are  two  of  the  feet  screws  which  support  the  entire  in- 
strument, and  by  which  the  column  carrying  the  telescope  is 
rendered  truly  vertical. 

C,  C  is  the  horizontal  circle,  12  inches  in  diameter,  graduated 
to  10',  and  reading  to  10",  by  means  of  its  vernier  and  micros- 
cope Y. 

B  is  therfcangent  screw  for  slow  motion. 

This  circle  serves  to  mark  the  position  of  the  meridian,  when 


154 


PRACTICAL   ASTRONOMY. 


LATITUDE.  155 

it  has  once  been  determined,  and  likewise  enables  the  observer 
to  turn  the  telescope  promptly  through  180°  in  azimuth. 

D  is  the  vertical  column  which  supports  the  telescope,  and 
about  which  the  telescope  turns  freely  in  azimuth. 

E  is  a  horizontal  axis,  to  one  end  of  which  is  attached  the 
telescope,  TT,  which  is  counterpoised  by  the  weight  "W,  at  the 
other  end. 

This  axis  is  hollow,  and  through  it  passes  the  light  of  the 
lamp,  H,  to  illumine  the  wires  of  the  telescope.  The  telescope 
has  a  focal  length  of  about  40  inches,  and  an  aperture  of  3  inches. 

L  is  a  level  resting  upon  the  horizontal  axis,  by  means  of 
which  the  column  D  is  rendered  truly  vertical. 

M  is  a  graduated  semicircle  attached  to  the  telescope,  and 
having  a  vernier,  N,  with  a  microscope.  This  semicircle  serves 
as  a  finder  for  setting  the  telescope  to  the  altitude  of  the  stars 
to  be  observed. 

S,  S  is  a  very  delicate  level  attached  to  the  semicircle. 

P  is  the  parallel  wire  micrometer  for  measuring  small  differ-     ^      *v 
ences  of  altitude,  having  three  fixed  vertical,  and  two  movable   *"* 
horizontal  wires.  m  *»  «• 

R  is  the  diagonal  eye-piece,  which  is  made  of  unusual  length, 
so  that  the  micrometer  may  not  interfere  with  the  observations. 
The  eye-pieces  employed  have  a  field  of  view  of  from  10X  to  15/. 

(182.)  Method  of  Observation. — Select  a  pair  of  stars,  the 
difference  of  whose  zenith  distances  does  not  exceed  a  convenient 
range  of  the  micrometer,  say  ten  minutes,  one  of  which  culmi- 
nates  to  the  north,  and  the  other  south  of  the  zenith.  Having  //0M*-vy**^ 
leveled  the  instrument,  set  the  telescope  to  an  altitude  midway 
between  the  two  stars,  and  bring  the  bubble  of  the  level  S  to  the 
middle  of  its  scale.  Bring  the  telescope  into  the  plane  of  the 
meridian  by  setting  the  vernier  of  the  horizontal  circle  to  the 
point  previously  determined.  As  the  first  star  enters  the  field 
of  view,  follow  its  image  with  one  of  the  horizontal  wires,  and 
bisect  it  at  the  instant  it  crosses  the  middle  vertical  wire.  Re- 
cord the  position  of  the  level  S,  noting  the  divisions  correspond- 
ing to  each  extremity  of  the  bubble.  Turn  the  telescope  180° 
in  azimuth,  being  careful  to  preserve  the  same  inclination  to  the 
horizon,  and  make  a  similar  observation  upon  the  second  star, 
bisecting  it  with  the  other  horizontal  wire. 


156  PRACTICAL   ASTRONOMY. 

A  comparison  of  the  readings  of  the  two  micrometer  screws 
will  give  the  difference  of  zenith  distance  of  the  two  stars,  which 
must  be  corrected  by  the  readings  of  the  level,  if  the  readings 
at  each  extremity  are  not  the  same  in  both  cases ;  and  also  for 
the  difference  of  the  refractions  of  the  two  stars. 

The  stars  should  be  so  selected  that  their  zenith  distances 
may  be  as  small  as  practicable,  and  should  in  no  case  exceed 
25  degrees. 

The  following  observations,  made  at  Mount  Independence, 
Maine,  one  of  the  coast  survey  stations,  September  25,  1849, 
will  illustrate  this  method.  The  pair  of  stars  employed  con- 
sisted of  Nos.  6983  and  6996  of  the  British  Association  cata- 
logue, whose  apparent  places  were 


No. 

Right  Ascension. 

Declination. 

6983 
6996 

20  10  50.05 
20  12  48.19 

47  15  40.70 
40  16  19.21 

The  formula  for  latitude  is 


0  9-  *m  Here  d.+ 4=87°  31'  59".91 ;  Zs-Zn  was  found,  by  observa- 
tion, equal  to  —  50x/.29.  Therefore  20=87 '°  317  9/x.62.  The 
observations  indicated  no  correction  for  level ;  and  the  correc- 
tion for  difference  of  refraction  was  —  0/x.02.  Hence  the  final 

latitude  is  43°  45'  34X/.80. 

-  -j 

September  27th,  the  same  stars  were  again  observed,  when 
d84-<?n  equaled  87°  32'  Ox/.40 ;  Zs-Zn  was  found  equal  to 
— 49/7.43.  The  correction  for  level  was  +0/x.90,  and  for  refrac- 
tion —  Ox/.02,  from  which  we  deduce  the  latitude,  43°  45X  35x/.92. 

(183.)  This  method  of  determining  latitude  possesses  the  fol- 
lowing advantages :  1.  It  eliminates  almost  entirely  the  effect 
of  atmospheric  refraction,  since  we  only  require  the  difference 
of  refraction  of  the  two  stars.  With  a  zenith  distance  of  25  de- 
grees, and  a  difference  of  altitude  of  247  between  the  two  stars, 
this  difference  of  refraction  does  not  exceed  half  a  second  of 
arc.  The  observations  are  generally  made  much  nearer  to  the 
zenith  than  25°,  and  the  difference  of  altitude  is  commonly  but 
a  few  minutes. 

2.  The  angular  measurements  required  are  made  by  means 
of  a  micrometer,  so  that  there  is  no  occasion  for  a  large  gradu- 


LATITUDE.  157 

ated  circle,  the  semicircle  attached  to  the  telescope  being  used 
merely  as  a  finder. 

The  chief  objection  to  this  method  is,  that  the  resulting  lati- 
tude is  affected  by  any  error  which  may  exist  in  the  assumed 
declinations  of  the  stars  employed,  and  we  are  generally  obliged 
to  make  our  selections  from  stars  whose  places  have  not  been 
determined  with  the  greatest  accuracy.  When  accurate  de- 
terminations of  the  stars  employed  can  be  obtained  with  the 
large  instruments  of  a  fixed  observatory,  this  objection  is  mostly 
obviated. 

SEVENTH    METHOD. 

(184.)  By  observations  with  a  transit  instrument  in  the 
prime  vertical. 

This  method  supposes  the  transit  instrument  to  be  placed 
with  its  supports  north  and  south,  so  that  the  telescope,  when 
directed  toward  the  horizon,  points  due  east  and  west.  We 
must  then  observe  the  passage  of  some  known  star  over  the 
same  wires  when  the  telescope  is  pointing  west.  From  these 
observations  we  may  determine  the  latitude  of  the  place,  or 
the  declination  of  the  star,  when  either  of  these  quantities  is 
known. 

Let  P  represent  the  pole  of  the  earth, 
Z  the  zenith  of  the  observer,  EZW  the 
prime  vertical,  which  is  also  the  line  de- 
scribed in  the  heavens  by  the  transit ; 
and  let  the  arc  SBSX  be  the  path  of  a 
star  which  culminates  a  little  south  of 
the  zenith.     Let  the  times  at  which  a  star  crosses  the  field  of 
the  transit  at  S  and  S'be  noted  ;  then  will  the  angle  SPSX,  which 
is  the  difference  of  those  times,  be  known.     Then,  in  the  right- 
angled  spherical  triangle  PZS,  by  Napier's  rule, 
R.  cos.  ZPS  =  tang.  PZ  cot.  PS. 

Put  ZPS=P— half  the  sidereal  interval  between  the  times 

of  east  and  west  transit ; 
(5=90°  —  PS = the  declination  of  the  star ; 
0=90°-PZ  =  the  latitude  of  the  place. 

Then 

cos.  P =cot.  0  tang.  6 ; 


158  PRACTICAL   ASTRONOMY. 

tang,  d 


which  is  the  same  as  given  in  Art.  148. 

Ex.  1.  On  the  16th  of  December,  1844,  the  transit  of  a  Lyree 
over  the  prime  vertical  of  Cambridge  was  observed  at  16h.  34m. 
47.3s.  ;  and  again  at  20h.  25m.  14.0s.  ;  the  declination  of  the 
star  being  38°  38'  42X/.05.  Required  the  latitude  of  the  obser- 
vatory. 

Here  P=lh.  55m.  13.35s.  in  time,  or  28°  48X  20x/.25  in  arc. 
tang,  d,  38°  38'  42X/.05  =  9.9028601 
cos.  P,  28°  48'  20//.25  =  9.9426327 
Latitude^  42°  22'  48x/.3  tang.  =  9.9602274 

Ex.  2.  On  the  4th  of  January,  1846,  the  transit  of  a  Lyrse 
over  the  prime  vertical  of  Washington  was  observed  at  18h.  27m. 
0.35s.  ;  and  again  at  19h.  28m.  1.0s.  ;  the  decimation  of  the  star 
being  38°  38'  427/.37.  Required  the  latitude  of  the  observatory. 

Here  P=0h.  30m.  30.325s.  in  time,  or  7°  37'  34".87  in  arc. 
tang.  d,  38°  38'  42".37=  9.9028615 
cos.  P,   7°  37X  34".87  =  9.9961414 
Latitude  =  38°  53'  37//.l  tang.  =  9.9067201 

(185.)  When  these  observations  are  made  for  the  determina- 
tion of  latitude,  it  is  best  to  select  a  star  which  culminates  but 
a  little  south  of  the  zenith,  as  the  same  error  in  the  observations 
will  have  less  influence  upon  the  result.  The  transit  instrument 
may  be  brought  nearly  into  the  prime  vertical,  by  computing 
the  time  when  a  star  which  culminates  several  degrees  south 
of  the  zenith  will  pass  the  prime  vertical.  The  formula 

cos.  P=oot.  0  tang.  6 

gives  the  hour  angle  between  the  meridian  and  the  time  of 
transit  over  the  prime  vertical.  The  right  ascension  of  the  star, 
minus  the  hour  angle,  gives  the  time  of  the  east  transit  ;  and 
the  right  ascension,  plus  the  hour  angle,  gives  the  time  of  west 
transit. 

(186.)  When  the  instrument  is  brought  nearly  into  the  prime 
vertical,  the  error  in  azimuth  may  be  determined  as  follows: 
Half  the  sum  of  the  times  of  transit  over  the  east  and  west  ver- 
ticals, gives  the  time  of  transit  over  the  meridian  of  the  instru- 
ment. This  result  should  be  equal  to  the  right  ascension  of  the 
star,  corrected  for  the  error  of  the  clock.  If  the  two  results  are 


LATITUDE.  159 

not  equal,  their  difference  shows  the  angle  which  the  meridian 
of  the  instrument  makes  with  the  true  meridian. 

If  the  plane  of  the  telescope  deviates  much  from  the  prime 
vertical,  the  co-latitude  deduced  will 
be  sensibly  too  small.  Suppose  the 
axis  deviates  to  the  east  of  north,  and 
that  the  telescope  describes  a  verti- 
cal circle,  passing  through  EXZWX; 
then  will  PZX,  which  bisects  SSX,  be 
the  co-latitude  which  results  from 
the  above  formula. 

The  correction  for  this  deviation  may  be  computed  as  follows : 
Take  the  half  sum  of  the  times  of  transit  over  the  east  and  west 
verticals,  correct  it  for  the  error  of  the  clock,  and  subtract  the 
result  from  the  star's  right  ascension.  The  difference  will  be 
the  angle  ZPZX.  Now,  from  the  right-angled  triangle  PZZX,  we 
have 

tang.  PZ  cos.  ZPZx=tang.  PZx=tang.  PS  x  cos.  SPZX, 

tang.  <5  x  cos.  ZPZX 
or  tang.  <p= —  — - . 

The  angle  ZPZX  is  the  same  for  all  stars,  and  it  is  better  to 
deduce  its  value  from  a  star  which  culminates  several  degrees 
south  of  the  zenith,  since  the  same  error  in  the  observations  will 
have  less  influence  upon  the  azimuth  deduced. 

(187.)  If  we  reverse  the  telescope  upon  its  supports,  any  error 
of  collimation  or  inequality  of  pivots  will  produce  exactly  a  con- 
trary effect  on  the  latitude.  Observations,  therefore,  of  two  stars 
on  the  same  day,  in  reversed  positions  of  the  telescope,  or  of  the 
same  star  on  following  days,  in  reversed  positions  of  the  tele- 
scope, will  correct  each  other,  and  the  mean  will  give  the  true  lat- 
itude, if  the  declination  of  the  star  is  accurately  known.  This 
is  one  of  the  best  methods  of  determining  the  latitude  with,  a 
portable  instrument. 

In  the  equation 

cos.  P=cot.  0  tang.  <5, 

either  </>  or  6  may  be  computed  when  the  other  quantity  is  known. 
Hence,  in  a  fixed  observatory,  when  the  latitude  is  well  de- 
termined, the  declinations  of  stars  may  be  determined  with  great 
precision  by  a  transit  instrument,  adjusted  to  the  prime  vertical. 


160  PRACTICAL   ASTRONOMY. 

But  to  accomplish  this  object  in  the  best  manner  requires  an 
instrument  of  a  peculiar  construction.  The  instrument  should 
admit  of  having  the  level  applied  to  it  while  the  telescope  is  in 
the  position  of  observation,  and  it  should  also  admit  of  being  re- 
versed with  ease  and  rapidity.  The  figure  on  the  opposite  page 
represents  the  instrument  used  for  this  purpose  at  the  "Washing- 
ton Observatory,  and  was  made  by  Pistor  and  Martins,  of  Berlin. 

(188.)  The  instrument  rests  on  a  block  of  granite,  MM,  6  feet 
5  inches  high,  3  feet  3  inches  from  east  to  west,  and  3  feet  7 
inches  from  north  to  south.  This  block  is  cut  so  as  to  form  two 
columns  4J  feet  high,  separated  by  a  cavity  which  contains  the 
reversing  apparatus. 

S  is  the  axis  of  the  instrument,  terminating  in  two  pivots, 
B,  B,  3.6  inches  in  diameter ;  to  one  of  which  is  attached  the 
telescope,  T,  to  the  other  the  cylinder,  U,  which  counterpoises 
the  telescope.  The  telescope  is  6£  feet  focal  length,  and  4.8 
inches  clear  aperture. 

V,  V  are  the  Y's  which  support  the  axis,  and  C,  C  are  friction 
rollers,  with  grooves  for  relieving  the  Y's.  They  are  regulated 
by  the  counterpoises  W,  "W,  all  of  which  are  carried  by  the  re- 
versing apparatus. 

The  axis,  S,  is  hollow,  and  contains  a  lever,  r,  one  end  of 
which  expands  into  a  fork,  and.  is  firmly  secured  at  x  to  each 
side  of  the  telescope  tube.  To  the  other  end  of  the  lever  is  at- 
tached the  counterpoise,  K,  which  transfers  the  weight  of  the 
telescope  to  that  part  of  the  pivot  which  rests  immediately  upon 
the  Y's.  A  similar  counterpoise  is  placed  on  the  other  side,  to 
produce  the  same  effect  with  reference  to  the  cylinder  U. 

Z  is  the  striding  level,  which  rests  permanently  upon  the  piv- 
ots B,  B  during  the  observations ;  and  L  is  a  mirror  for  illumina- 
ting the  level  divisions  by  means  of  a  lamp.  The  level  tube  is 
protected  by  a  glass  case,  Gr,  and  there  is  a  cross  level  at  h. 

About  the  middle  of  the  axis,  at  d,  is  a  clamp  for  slow  mo- 
tion of  the  telescope,  and  a  screw,  with  a  Hook's  joint,  at  E. 

The  reversing  apparatus,  P,  P,  turns  on  an  inverted  cone, 
working  in  the  hollow  cylinder,  R,  and  is  strengthened  by  the 
cross  iron  bars,  #,  a,  a,  which  are  supported  by  the  flat  iron  bars, 
b,b. 

H  is  a  crank  which  turns  a  cog-wheel  at  N,  which,  by  means 


LATITUDE. 


161 


T, 


162  PRACTICAL   ASTRONOMY. 

of  a  screw,  lifts  the  hollow  cylinder,  R,  and,  by  means  of  the 
forks,  /,  /,  lifts  the  horizontal  axis  until  the  pivots,  B,  B,  are 
sufficiently  high  to  clear  the  Y's.  The  telescope  is  then  turned 
to  a  zenith  distance  of  about  45°,  and  is  revolved  to  the  other 
side  of  the  pier.  It  is  prevented  from  going  too  far  by  the  arm 
F,  which  is  so  adjusted  as  to  strike  the  pin  D,  when  the  tele- 
scope is  exactly  over  the  Y's. 

n  is  a  finding  circle  for  setting  the  telescope  upon  a  star. 

J  is  the  handle  of  a  screw,  which  moves  a  slide  at  0  for  reg- 
ulating the  illumination  of  the  wires. 

t  is  the  micrometer  head  and  screw  moving  the  micrometer 
wire. 

p  is  a  lever  which  carries  the  eye-piece  across  the  field. 

In  the  eye-piece  of  the  telescope  are  inserted  two  horizontal 
and  parallel  threads,  distant  V  from  each  other ;  and  also  15 
fixed  vertical  lines,  with  one  movable  one.  The  transits  over 
the  vertical  lines  are  designed  to  be  observed  midway  between 
the  two  horizontal  lines. 

(189.)  Mode  of  observation. 

Having  determined  the  error  of  level  of  the  axis,  direct  the 
telescope  to  a  star  while  it  is  yet  north  of  the  eastern  prime 
vertical,  and  observe  the  transit  of  the  star  over  each  of  the 
wires  preceding  the  middle  of  the  field ;  the  altitude  of  the  tel- 
escope being  continually  changed,  so  that  the  oblique  transit  may 
be  observed  over  the  centre  of  each  wire.  When  the  star  has 
passed  the  wire  next  before  the  middle,  reverse  the  axis,  by 
which  means  the  telescope  will  be  carried  to  the  opposite  side 
of  the  pier,  and  observe  the  passage  of  the  star,  now  on  the  south 
side  of  the  eastern  prime  vertical,  over  the  same  wires  as  before, 
but  in  the  opposite  order.  Determine  again  the  error  of  level 
of  the  axis.  When  the  star  is  approaching  the  western  prime 
vertical  from  the  south,  the  instrument  being  still  in  its  second 
position,  ascertain  again  the  error  of  level  of  the  axis.  Again 
observe  the  transit  of  the  star  over  the  first  seven  wires  preceding 
the  middle  of  the  field ;  reverse  the  instrument  to  its  first  posi- 
tion, and  observe  the  transit  of  the  star,  now  on  the  north  side 
of  the  western  prime  vertical,  over  the  same  wires.  Finally, 
ascertain  the  error  of  level  of  the  axis  in  the  last  position. 

The  following  observations  were  made  by  Struve,  with  the 


LATITUDE. 


163 


prime  vertical  transit  of  the  Pulkova  Observatory.     The  num- 
bers in  the  last  column  are  read  from  below,  upward. 

January  15,  1842.     o  Draconis. 


EAST    VERTICAL. 

WEST    VERTICAL. 

Telescope  S. 

Telescope  S. 

Wires. 

h.      m.         s. 

ft.     m.         s. 

I. 

17  54  30.7 

19  42  51.4 

II. 

55     8.65 

42  13.65 

III. 

55  44.4 

41  38.0 

IV. 

56  22.25 

40  59.85 

V. 

57     0.6 

40  21.7 

VI. 

57  40.9 

39  41.4 

VII. 

17  58  19.5 

19  39     2.7 

Telescope  N. 

Telescope  N. 

VII. 

18     1     4.0 

19  36  17.85 

VI. 

1  45.5 

35  37.0 

V. 

2  29.8 

34  52.35 

IV. 

3  12.7 

34     9.3 

III. 

3  57.6 

33  24.7 

II. 

4  39.8 

32  42.1 

I. 

18     5  26.35 

19  31  55.6 

Level  =  +0/x.687 

Level  =  +Ox/.923 

(190.)  The  reduction  of  the  observations  is  made  as  follows, 
each  wire  being  treated  separately. 

Let  NESW  represent  the  horizon, 
NS  the  meridian,  E  W  the  prime  vert- 
ical, P  the  pole,  and  A  the  place  of  the 
star  at  its  transit  over  one  of  the  wires 
of  the  telescope.  Join  PA  and  NA 
by  arcs  of  great  circles.  The  projec- 
tion of  each  wire  on  the  sky  is  a  small 
circle,  whose  pole  is  the  north  point, 
N,  of  the  horizon.  If  c  represent  the 
angular  distance  of  one  of  the  wires  from  the  line  of  collimation, 
90°  —  c  will  be  the  radius  NA  of  the  small  circle,  when  the  star 
is  seen  on  it,  north  of  the  prime  vertical,  and  90°  +  c  when  the 
star  is  south  of  the  prime  vertical. 

In  the  triangle  PNA,  by  Trig.,  Art.  225,  we  have 
cos.  NA=cos.  NP  cos.  PA + sin.  NP  sin.  PA  cos.  NPA. 

Let     0=:NP  the  latitude  of  the  place  ; 


164 


PRACTICAL   ASTRONOMY. 


<S-90°  —  PA=the  star's  declination; 
t,  t'  —  the  hour  angles  SPA  from  the  meridian,  at  the  two 
observations  over  the  same  wire,  in  the  direct  and 
reversed  positions  of  the  axis. 
Then,  when  the  star  is  north  of  the  prime  vertical, 

cos.  (90°  —  c)=sin.  c  =  cos.  </>  sin.  6—  sin.  0  cos.  6  cos.  t; 
and,  when  the  star  is  south  of  the  prime  vertical, 

cos.   90°  +  c=:  —  sin.  c  =  cos.  (>  sin.  6  —  sin.     cos.  6  cos.  t'  '. 


/ox 
(2.) 


Adding  these  two  equations,  we  obtain 

0  =  2  sin.  d  cos.  0  —  cos.  d  sin.  <£  (cos.  t  -\-cos.  £'), 
or,  Trig.,  Art.  75, 


tang.  6  cot.  </>  = 


cos. 


V  t' 

=  cos. 


cos. 


t'  —  t 


This  formula  will  furnish  the  decimation  when  the  latitude 
is  known,  or  the  latitude  when  the  declination  is  known.  The 
latitude  of  the  Pulkova  instrument  is  59°  46'  18/x.  t'  represents 
half  the  interval  between  the  first  transit  east  and  the  second 
transit  west  ;  and  t  is  half  the  interval  between  the  second  tran- 
sit east  and  the  first  transit  west. 

(191.)  The  following  is  Struve's  reduction  of  the  preceding 
observations,  a  correction  of  +  0.09s.  being  applied  to  the  inter- 
val W.  —  E.  for  rate  of  clock. 


Wire  I.    Wire  II. 

Wire  III.  |  Wire  IV. 

Wire  V. 

Wire  VI.  |Wire  VII. 

W.-E.{»f 

4(*'+0 
Kf-0 

COS.  i(<'  +  0 

cos.  \(t'-t) 
tang.  $ 
tang,  d 

6 

h.  m.   s.  1  m.   s. 
1  48  20.79  47  5.09 
1  26  29.34  28  2.39 
0  48  42.53  48  46.87 
0  5  27.86  4  45.67 

m.   s. 
45  53.69 
29  27.19 
48  50.22 
4  6.62 

m.   s. 
44  37.69 
30  56.69 
48  53.60 
3  25.25 

TO.    S. 

43  21.19 
32  22.64 
48  55.96 
2  44.64 

TO.    S. 

42  0.59 
33  51.59 

48  58.05 
2  2.25 

TO.    S. 

40  43.29 
35  13.94 
48  59.31 
1  22.33 

9.9901167 
9.9998765 
0.2345728 

0871 
9063 

5728 

0642 
9301 

5728 

0411 
9516 

5728 

0250 
9688 

5728 

0107 

9828 
5728 

0020 
9922 

5728 

0.2245660 
5°9  11  39'.00 

5662 
39.04 

5671 
39.23 

5655 
38.90 

5666 
39.'l2 

5663 
39.06 

5670 
39.21 

The  mean  error  of  level  of  the  instrument  may  be  applied  to 
0,  or  we  may  apply  a  correction  to  the  declination  obtained  with 
a  constant  value  of  0.  If  the  inclination  of  the  axis  be  denoted 
by  I,  which  is  the  mean  of  the  two  inclinations,  telescope  IN"  and 
telescope  S,  then  0  +  1  should  be  used  in  place  of  </>,  in  formula 
(2),  Art.  190.  Now,  by  formula  (1),  Art.  184,  we  have 
tang.  6= tang.  0  cos.  P. 

By  differentiating,  supposing  P  constant,  we  obtain 


LATITUDE. 


165 


dd  sec.  26=fl(f)  sec.  20  cos.  P. 


or 


Hence 
dd    d6SGC'2 

0  tang. 

6  ,  cos.  c 

J  sin.  6 

sec.   i 

d  tang. 
•    / 

0  r  cos.  (/ 
7,  sin.  26T 

In  —  1 

>  sin.  0 

, ,  sin.  2<5 
-^sin.^' 


sin. 


In  the  preceding  observations  the  mean  inclination  of  the  axis 
was  +0X/.805. 

The  mean  value  of  6  =59°  \V  39".071 

Correction  for  inclination  of  axis  -j-0x/.814 

Observed  declination  =59°  if  ~39".88o 

(192.)  The  declination  thus  found 
is  not  correct,  unless  the  telescope  is 
truly  adjusted  to  the  prime  vertical. 
Suppose  there  is  an  error  in  the  azi- 
muth of  the  instrument  equal  to  a  or 
90°  -  PZZX;  then,  in  the  triangle  PZZ7, 


•n     cot.  PZZ' 
tans*  P  =  ^ 

cos.  PZ 


tang,  a 


sin.  0 
If  the  error  in  azimuth  be  small,  we  may  assume 

p^     a 

sin.  0' 

which  represents  the  angle  at  the  pole,  between  the  true  me- 
ridian and  the  meridian  of  the  instrument.  The  instant  of  the 
star's  passage  over  the  meridian  of  the  instrument  is  equal  to  the 
half  sum  of  the  east  and  west  transits.  Thus,  in  the  preceding 
observation,  we  have 

Wires.  Telescope  S. 

I.  18h.  48m.  41.10s. 

II.  41.15s. 

III.  41.20s. 

41.05s. 
41.15s. 
41.15s. 
41.10s. 


IV. 

V. 

VI. 

VII. 


Telescope  N. 

18h.  48m.  40.93s. 
41.25s. 
41.07s. 
41.00s. 
41.15s. 
40.95s. 
40.97s. 


41.05s. 


Mean,  18h.  48m.  41.13s. 

Mean,  ISh.  48m.  41.09s. 

The  instant  of  meridian  passage  requires  a  small  correction 
for  the  difference  of  inclinations  of  the  axis  in  the  two  verticals. 


166  PRACTICAL    ASTRONOMY. 

This  correction  in  the  present  case  amounts  to  —  0.08s. ;  and 

hence  the  true  time  of  meridian  passage  by  the  instrument  is 

18h.  48m.  41.01s. 

The  star's  right  ascension,  corrected  for  error  of  the  clock,  was 
18h.  48m.  41.86s. 

Hence  P  —  —0.85s.  in  time,  is  the  angle  of  the  two  meridians. 

For  the  azimuth  of  the  axis  of  rotation,  reckoned  from  the 
south  round  by  the  west, 

&=15P  sin.  </>,  in  arc. 

In  the  present  case, 

a——  llx/.0,  in  arc. 

The  effect  of  this  small  azimuthal  error  upon  the  declination 
is  inappreciable. 

It  is  the  opinion  of  Struve  that,  with  this  instrument,  changes 
in  the  apparent  decimation  of  zenith  stars,  amounting  to  a  small 
fraction  of  a  second,  may  be  detected.  This  instrument  may 
therefore  be  employed  to  determine  the  aberration  of  light,  and 
the  annual  parallax  of  zenith  stars.  The  bright  star  a  Lyrse 
culminates  about  15'  south  of  the  zenith  of  Washington  Observa- 
tory, and  this  star  has  been  observed  by  Professor  Hubbard  with 
great  care,  for  the  purpose  of  determining  its  annual  parallax, 
which,  according  to  the  Pulkova  observations,  amounts  to  about 


CHAPTER  VII. 

ECLIPTIC. 

(193.)  WHEN  an  observer  has  obtained  the  latitude  of  his 
station,  he  is  prepared  with  an  astronomical  circle  to  determine 
the  apparent  declinations  of  the  heavenly  bodies.  For  the  eleva- 
tion of  the  equator,  EH,  is  the  com- 
plement of  PO,  the  elevation  of  the 
pole  ;  and  if  from  SH,  the  altitude 
of  a  star,  we  subtract  EH,  the  ele- 
vation of  the  equator,  we  shall  ob- 
tain the  star's  declination.  This 
rule  will  hold  for  all  the  heavenly  bodies  at  their  upper  culmin- 
ation, if  we  measure  their  altitude  from  the  south  horizon.  Or, 
if  we  represent  the  latitude  by  </>,  and  the  zenith  distance  by 
Z,  when  a  body  culminates  south  of  the  zenith,  we  have 

<5=</>-Z. 

If  it  culminate  north  of  the  zenith,  and  above  the  pole, 
rf=0+Z. 

If  it  culminate  north  of  the  zenith,  and  below  the  pole, 


(194.)  If  the  declination  of  the  sun  be  observed  during  a 
whole  year,  whenever  it  passes  the  meridian,  upon  comparing 
the  results  it  will  be  found  that,  on  the  22d  of  December,  the 
declination  has  its  greatest  value  on  the  southern  side  of  the 
equator  ;  that  it  diminishes-till  the  21st  of  March,  when  the 
declination  is  exactly  or  nearly  zero  ;  and  that  it  afterward  in- 
creases on  the  northern  side  of  the  equator  till  June  21.  From 
this  time  the  declination  diminishes  till  the  23d  of  September, 
when  it  is  again  zero,  and  increases  again  on  the  southern  side 
of  the  equator  till  the  22d  of  December. 

The  greatest  observed  northern  and  southern  declinations  of 
the  sun  constitute  approximate  values  of  the  angles  at  which 
the  plane  of  the  ecliptic  and  the  plane  of  the  equator  intersect 
each  other  j  and  the  times  at  which  the  declinations  are*  nearly 


168  PRACTICAL   ASTRONOMY. 

zero,  are  the  approximate  times  when  the  sun,  in  ascending  and 
descending,  crosses  the  plane  of  the  earth's  equator ;  but  as  the 
observations  are  only  made  at  the  instants  of  apparent  noon  at 
the  station,  it  is  not  probable  that  the  greatest  or  least  declination 
will  take  place  precisely  at  the  instant  of  observation  ;  and  there- 
fore a  computation  must  be  made  to  obtain  these  elements  with 
sufficient  accuracy. 

Right  ascension  is  reckoned  from  the  vernal  equinox ;  and  a 
clock  regulated  to  exact  sidereal  time  should  indicate  Oh.  Om.  Os. 
when  the  vernal  equinox  is  passing  the  meridian. 

PROBLEM. 

(195.)  To  find  the  position  of  the  equinoctial  points. 
Observe  the  altitude  of  the  sun  when  on  the  meridian  upon 
the  day  which  precedes  and  the  day  which  follows  the  equinox. 
These  altitudes,  corrected  for  refraction  and  parallax,  will  fur- 
nish the  declinations  d  and  6',  one 
south  and  the  other  north.  Let 
T  represent  the  interval  between 
the  observations,  expressed  in  si- 
dereal time.  Let  A  be  the  place 
of  the  equinox,  BBX  the  equator,  SSX  the  ecliptic,  S  and  Sx  the 
places  of  the  sun  on  two  successive  days,  one  preceding  and  the 
other  following  the  equinox ;  also,  let  BS  and  B'SX  be  the  ob- 
served declinations.  Then,  suppose  the  motion  in  declination 
and  right  ascension  to  be  uniform  at  this  time,  as  they  are  very 
nearly,  we  shall  have 

BS  +  B'S'rBS.-rBB'iBA, 
or       6+ 6' :  6 : :  T  —  24h. :  diff.  right  asc.  between  B  and  A. 

Ex.  1.  On  the  20th  of  March,  1851,  the  sun's  decimation  at 
noon  was  observed  at  Greenwich  to  be  16'  33x/.29  S.,  and  March 
21st  it  was  7/  7/x.54  N. ;  also  the  sidereal  interval  between  the 
observations  was  24h.  3m.  38.23s.  What  was  the  sun's  right 
ascension  at  noon,  March  21st  ? 

In  this  case  we  shall  have  the  proportion 

16'  33x/.29  +  7x  7//.54:7/  7//.54::3m.  38.23s. :  65.67s. 
Therefore  the   sun's  right  ascension  at  noon,  March  21st, 
was 

Oh.  1m.  5.67s. 


ECLIPTIC.  169 

Ex.  2.  On  the  22d  of  September,  1851,  at  noon,  the  sun's 
declination  observed  at  Grreenwich  was  3X  51/x.53  N.,  and  on  the 
23d  it  was  19X  33/x.76  S. ;  also  the  sidereal  interval  of  the  tran- 
sits was  24h.  3rn.  36.13s.  "What  was  the  sun's  right  ascension 
at  noon,  September  23d  ? 

Ans.  12h.  3m.  0.52s. 

Ex.  3.  On  the  22d  of  September,  1846,  at  noon,  the  sun's 
declination,  observed  at  "Washington,  was  17'  2/x.80  N.,  and  on 
the  23d  it  was  6/  21". 56  S. ;  also,  the  sidereal  interval  of  the 
transits  was  24h.  3m.  35.50s.  What  was  the  sun's  right  ascen- 
sion at  the  second  observation  ? 

Ans.  12h.  Om.  58.55s. 

(196.)  These  computations  should  be  made  both  for  March 
and  September,  when  the  sun  crosses  the  equator.  If  the  side- 
real clock  were  correct,  it  would  be  found  to  indicate  12  hours 
when  the  sun  is  on  the  meridian  at  the  autumnal  equinox ;  from 
which  we  infer  that  the  two  equinoctial  points  are  distant  from 
one  another  180  degrees.  If  now  wTe  observe  some  star  which 
passes  the  meridian  about  the  same  time  with  the  vernal  equi- 
nox (as,  for  example,  a  Andromedse),  its  right  ascension  will  be 
known ;  and  having  settled  the  right  ascension  of  one  star,  the 
right  ascension  of  other  stars  may  thence  be  deduced.  Thus, 
taking  the  apparent  right  ascension  of  a  Andromedse  on  January 
31,  1853,  to  be  Oh.  Om.  46.10s.,  let  the  index  of  the  clock  be  set 
to  that  time  when  a  Andromedse  is  on  the  meridional  wire  of  the 
transit  telescope.  The  clock,  if  it  goes  correctly,  will  denote  the 
right  ascension  of  other  stars  when  they  are  bisected  by  the  me- 
ridional wire.  Thus,  on  the  above  day, 

Aldebaran  passing  the  meridional  wire  at  4h.  27m.  29.19s. 

Capella  "         "  "  "     "  5h.    5m.  50.26s. 

Rigel  "         "  "  "     "  5h.    7m.  28.58s. 

Sirius  "         "  "  "     "  6h.  38m.  40.82s. 

these  times  would  be  the  apparent  right  ascensions  of  those 
stars. 

The  star  selected  by  any  astronomer  to  regulate  the  right  as- 
censions of  other  stars  is  called  his  fundamental  star.  Dr.  Mas- 
kelyne,  at  the  Grreenwich  Observatory,  employed  a  Aquilea  for 
this  purpose  ;  while,  at  the  Washington  Observatory,  a  Androm- 
edse is  employed. 


170  PRACTICAL   ASTRONOMY. 

(197.)  When  a  number  of  stars  have  had  their  right  ascen- 
sions determined  by  referring  them  to  some  fundamental  star, 
they  will  all  be  charged  with  the  error  which  may  happen  to 
belong  to  this  star ;  and  it  is  an  object  of  the  utmost  importance 
to  ascertain  the  existence  and  quantity  of  such  error.  The  dif- 
ficulty lies  in  determining  accurately  the  position  of  the  first 
point  of  Aries,  from  which  the  right  ascensions  of  all  the  stars 
are  counted.  The  course  pursued,  therefore,  by  astronomers,  is 
first  to  find  the  sun's  right  ascension,  by  comparing  the  transit 
of  his  centre  with  the  transit  of  the  fundamental  star,  or  with 
the  transits  of  several  principal  stars,  related  to  it  by  known  dif- 
ferences ;  and,  secondly,  to  compute  from  his  observed  decima- 
tion the  right  ascension  belonging  to  the  moment  of  the  meridian 
passage.  These  operations  should  be  performed  on  several  days, 
near  both  the  vernal  and  autumnal  equinox.  The  right  ascen- 
sions derived  from  a  comparison  with  the  stars  should  agree 
with  those  derived  from  the  observed  declinations  of  the  sun. 
If  there  be  a  constant  difference,  this  will  be  the  correction  to 
be  applied  to  the  assumed  right  ascension  of  the  fundamental 
star.  The  sun's  right  ascension  is  deduced  from  his  declination 
in  the  following  manner  : 

Let  AC  represent  a  part  of  the 
equator,  AD  a  part  of  the  ecliptic, 
and  A  be  the  first  point  of  Aries. 
Suppose  the  sun  to  be  at  S,  and 
draw  SB  perpendicular  to  AC  ;  then 
AB  will  be  the  right  ascension  of  the  sun,  and  SB  his  declina- 
tion. 

But,  by  Napier's  rule, 

rad.  x  sin.  AB  =  cotang.  SAB  x  tang.  SB  ; 
that  is, 

sin.  R.  A.  =  cotang.  obliquity  x  tang.  dec. 
Or,  representing  the  sun's  declination  by  <5,  and  the  obliquity 
of  the  ecliptic  by  «,  we  have 

-^  tang.  6 

sin.  R.  A.= 2 — . 

tang,  w 

Ex.  1.  The  following  observations  of  the  sun's  centre  were 
made  at  Greenwich  in  1851 : 


E  CLIPTIC. 


171 


Date. 

Sun's  R.  A.  observed. 

Sun's  Dec.  observed.    . 

Sept.  15 
16 
21 
22 
23 

A.       m.            *. 

11  34  16.15 
37  51.22 
55  48.55 
59  24.19 
12     3     0.32 

2  47     0.18  N. 
2  23  50.70  N. 
0  27  15.35  N. 
0     3  51.53  N. 
0  19  33.76  S. 

It  is  required  to  find  the  mean  correction  of  the  right  ascen- 
sions, the  obliquity  of  the  ecliptic  being  23°  27'  28X/.15. 
The  computation  for  September  15  is  as  follows : 

tang,  d,    2°  47'    0".18  = 8.6867922 
tang.w,  23°  27X  28/x.15  =  9.6374270 
sin.  R.  A.,  llh.  34m.  16.13s.  =  9.0193652 
The  observed  right  ascension  was  llh.  34m.  16.15s. 
Error  of  the  observed  R.  A.  +  0.02s. 
In  the  same  way  the  2d  observation  gives  +  0.03s. 
"  "  3d          "  "     -0.19s. 

"  "  4th         "  "     -0.24s. 

"  "  5th        «  "     -0.01s. 

The  mean  =-  0.08s. 

That  is,  the  observed  right  ascensions  appear  to  be  too  small  by 
0.08s. 

Similar  observations  should  be  made  at  each  equinox  every 
year,  until  it  appears  that  no  further  correction  is  required. 

Ex.  2.  The  following  observations  of  the  sun's  centre  were 
made  at  Washington  in  1846 : 


Date. 

Sun's  R.  A. 

Sun's  Dec. 

Sept.  16 
21 
22 
23 
25 
28 

11  35  49.40 
11  53  46.89 
11  57  22.79 
12  0  58.29 
12  8  10.58 
12  18  59.37 

+2  36  56.46 
+  0  40  29.52 
+  0  17  2.80 
'-0  6  18.71 
-0  53  11.50 
-2  3  28.11 

It  is  required  to  find  the  mean  correction  of  the  right  ascen- 
sions, the  obliquity  of  the  ecliptic  being  23°  27'  25/x.88. 

Ans.  +  0,06s. 


172 


PRACTICAL   ASTRONOMY. 


PROBLEM. 

(198.)  To  find  the  obliquity  of  the  ecliptic. 
Observe  the  right  ascension  and  declination  of  the  sun  near 
one  of  the  solstices.  If  the  sun  were  exactly  at  the  solstice  at 
one  of  the  observations,  the  observed  declination  would  be  the 
obliquity  required.  But  as  such  a  coincidence  can  seldom  hap- 
pen, it  is  customary  to  take  observations  on  several  days  both 
before  and  after  the  solstice,  and  compute  the  reduction  to  the 
solstice.  This  may  be  done  in  the  following  manner : 

Let  AB  represent  the  equa- 
tor, AS  the  ecliptic,  A  the  vernal 
equinox,  S  the  solstice,  and  Sx 
the  place  of  the  sun  near  the 
solstice.      Let  fall  the  perpen- 
A    dicular  S'B'  upon  the  equator. 
Then,  by  Napier's  rule, 

R.  sin.  ABx=tang.  S/B/  cot.  BAS. 

Put  d=the  observed  declination;  7i  =  BB/  =  6h.— the  sun's 
right  ascension ;  w=the  obliquity  of  the  ecliptic ;  and  a;  — the  re- 
quired correction  to  obtain  the  declination  at  the  solstice.     Then 
sin.  ABx=tang.  6  cot.  o>, 

tang.  <5 

cos  h  =  —     — . 
tang,  cj 

By  Trig.,  Art.  76, 


B' 


or 


sn. 
sin. 


—  d)_tang.  w—  tang.  6_ 
tang.  w-f  tang.  6 


1- 


1+ 


tang.  6 

tang,  a)  _  1  —  cos.  li 

tang.  (5     1  +  cos.  h 


tang.  w 
-^     Trig.,  Art.  74,  =tang.  2JA; 


that  is, 


2  cos. 

sin.  (w  —  (5)  =  tang.  z£h  sin.  (w-fd). 
When  the  required  correction  is  small,  we  may  put  o>  —  6  for 
sin.  (w—  d),  and  dividing  by  sin.  l/x,  to  have  x  expressed  in  sec- 
onds, we  obtain 


which  is  the  correction  in  seconds  to  be  added  to  the  observed 
declination,  to  obtain  the  obliquity. 


ECLIPTIC.  173 

Ex.  1.  In  June,  1851,  the  following  observations  were  made 
at  Greenwich : 

June  17.  Sun's  R.  A.  5h.  40m.  59.17s.       Dec.  23°  23X    7x/.57 

«      19.  "       5     49      18.63  «  26     4  .87 

«      21.  «       5     57      37.76  "  27  22  .56 

"      26.  "       6     18      25.27  "  23  21  .25 

«      27.  "       6     22      34.12  «  21   18  .59 

«      28.  "       6     26      43.59  «  18  53  .14 

«      30.  "       6     35        1.20  «  12  47  .16 

It  is  required  to  determine  the  obliquity  of  the  ecliptic. 

Assume  for  the  obliquity  the  greatest  observed  declination,  or 

23°  27"  22".56.     Then,  for  June  17th>  the  reduction  will  be  as 

follows : 

A  =  19m.  0.83s. =4°  45X  12/x.45;  JA  =  2°  22'  36". 2. 
By  formula  (1), 

tang.  P  =  8.618105 
8.618105 

sin.  46°  50'  30".13  =  9.863005 

cosec.  1"  =  5.314425 

Correction,  259/x.20  = 2.413640 

This  correction  being  added  to  the  observed  declination,  23° 
23'  7/x.57,  gives 

The  obliquity  of  the  ecliptic =23°  27X  26/x.77 

In  like  manner  the  2d  observation  gives  26  .80 

".        "  3d         "  "  26  .59 

"         "  4th        "  «  24  .55 

"         "  5th        "  "  23  .78 

6th        «  "  25  .17 

"         "  7th       ,"  "  26  .39 

The  mean  is  23°  27X  25".72 

(199.)  It  may  be  thought  that  this  method  involves  a  vicious 
principle,  inasmuch  as  it  requires  a  knowledge  of  w  to  enable 
us  to  find  the  value  of  w.  But  it  will  be  noticed  that  only  an 
approximate  knowledge  of  G>  is  required  to  furnish  a  very  accu- 
rate value  of  the  correction  x.  In  reducing  the  preceding  ob- 
servation of  June  17,  an  error  of  one  minute  in  the  assumed 
value  of  w  will  occasion  an  error  of  less  than  one  tenth  of  a  sec- 
ond in  the  computed  reduction. 


174 


PRACTICAL   ASTRONOMY. 


This  is  but  one  example  of  a  very  common  case  in  astronomy, 
in  which  we  employ  an  approximate  value  of  an  unknown  quan- 
tity to  obtain  a  more  accurate  determination. 

Ex.  2.  The  following  observations  were  made  at  Greenwich 
in  1850 : 

June  17.  Sun's  R.  A.  5h.  41m.  59.36s.     Dec.  23°  23'  29".18 


7  .16 

0  .73 

23  .24 

17  .83 
.95 

.84 


54 

38 
55 


"    18.            "        5     46        8.94  "           25 

"    20.             "        5     54      27.74  «            27 

«    21.             "        5      58      37.40  «           27 

"    22.             "62     46.82  "            27 

"    24.            "        6      11       5.83  «           25 

"    25.            «       6      15     14.88  "           24 

"    26.             "        6      19      24.40  «            22   55  .04 
Required  the  obliquity  of  the  ecliptic. 

Ans.  23°  27/  23//.88. 

Ex.  3.  The  following  observations  were  made  at  "Washington 
in  1846 : 

Sun's  R.  A.  17h.  44m.  37.21s.  Dec.  23°  24'  32/x.69 

«         17     57      56.81  "            27   20  .43 

"         18       2      23.51  «            27   19  .64 

"         18       6      50.17  «            26   49  .82 


Dec.  18. 
"  21. 
"  22. 
«  23. 


Required  the  obliquity  of  the  ecliptic. 


Ans.  23°  27'  23//.20. 


PROBLEM. 


(200.)   To  find  the  longitude  and  latitude  of  a  star,  when 
its  right  ascension  and  declination  are  known. 

Let  P  represent  the  pole  of  the 
equator,  E  the  pole  of  the  eclip- 
tic, C  the  first  point  of  Aries, 
PSPX  an  hour  circle  passing 
through  the  star  S,  and  ESE'  a 
circle  of  latitude  passing  through 
the  same  star.  Then  AEBE' 
represents  the  solstitial  colure, 
EP  represents  the  obliquity  of 
the  ecliptic,  PS  the  polar  dis- 
tance of  the  star,  ES  its  co-lati- 
tude ;  SPB  is  the  complement  of  its  right  ascension,  and  SEE 


ECLIPTIC.  175 

is  the  complement  of  its  longitude.  Draw  SM  perpendicular 
to  PB.  Represent  PM  by  a  ;  also  represent  the  longitude  of  the 
star  S  by  L,  its  latitude  by  /,  and  the  obliquity  of  the  ecliptic 
by  to. 

Now,  by  Napier's  rule,  we  have 

R.  cos.  SPM=tang.  PM  cot.  PS; 
that  is,  sin.  R.  A.  =  tang.  a  tang.  Dec., 

or  tang.  «=sin.  R.  A.  cot.  Dec  ......  (A) 

Also,  EM  =  EP  +  PM  =  a+w. 

Again,  Trig.,  Art.  216,  Cor.  3, 

sin.  EM  :  sin.  PM  ::  tang.  SPM  :  tang.  SEM  ; 
that  is, 

sin.  (a+  <y)  :  sin.  a  :  :  cot.  R.  A.  :  cot.  L  :  :  tang.  L  :  tang.  R.  A., 


T      tang.  R.A.  sin.  1X 

or  tang.  L  =  —  —  ^  —  -  —  -    .....  (1) 

sin.  a 

Also,  tang.  EM  cot.  ES  =  R.  cos.  SEM  ; 

that  is,  tang.  l=cot.  (a+u)  sin.  L    ......  (2) 

Also,  Trig.,  Art.  216, 

cos.  PM  :  cos.  EM  ::  cos.  PS  :  cos.  ES  ; 

,  ,       .  .      ,     cos.  (a+  G>)  sin.  Dec.  Q. 

that  is,  sin.  /=—  -  .......  (3) 

cos.  a 

And  R.  cos.  SEP=tang.  EM  cot.  ES; 

that  is,  sin.  L=tang.  (o  +  w)  tang.  /    ......  (4) 

Ex.  1.  On  the  1st  of  January,  1851,  the  R.  A.  of  Capella  was 
5h.  5m.  42.03s.,  and  its  Dec.  45°  50'  22"  A  N.  ;  required  its  lati- 
tude and  longitude,  the  obliquity  of  the  ecliptic  being  23°  27' 
25"A7. 

By  equation  (A), 

R.  A.  76°  25'  30/x.45     sin.  =  9.9876948 
Dec.  45°  5(K  22"  A       cot.  =  9.9872707 
«=43°  20X  58/x.31  tang.  =  9.9749<?55 
6>  =23°  27X  25/x.47 
a  +6)  =66°  48/23//.78 

By  equation  (1), 

tang.  R.A.  =  0.6171524 

sin.    a  +w  =  9.9634009 


cosec. 


=  79°  46X  40/x.93  tang.  =  0.7439461 


176 


PRACTICAL   ASTRONOMY. 


By  equation  (2), 


cot.  («  +  w)  =  9.6319144 
.  sin.  L  =  9.9930515 
/=22°  51'  48X/.14  tans.  =9.6249659 


By  equation  (3), 


cos.  (a  +  w)  =  9.5953154 
sin.  Dec.  =  9.8557564 
sec.  a  =  01383583 
/=22°  51'  48X/14  sin.  =  9.5894301 

By  equation  (4),  tang,  (a  +  w)  =  0.3680856 

tang.  /.  =  9.6249659 
L  =  79°  46'  41X/.00  sin.  =  9.9930515 

Formulas  (3)  and  (4)  give  nearly  the  same  result  as  formu- 
las (1)  and  (2).  Formulas  (1)  and  (2)  are,  however,  to  be  pre- 
ferred, because  the  tangents  vary  more  rapidly  than  the  sines, 
especially  near  90°. 

Formulas  which  furnish  the  value  of  an  unknown  quantity 
by  means  of  its  tangent  or  cotangent,  are  generally  more  accu- 
rate than  those  which  furnish  it  by  means  of  its  sine  or  cosine. 

Ex.  2.  On  the  1st  of  January,  1851,  the  R.  A.  of  Regulus 
was  lOh.  Om.  25.87s.,  and  its  Dec.  12°  41X  32x/.7  N.  Required 
its  latitude  and  longitude,  the  obliquity  of  the  ecliptic  being  23° 
27'  25x/.47.  Ans.  Latitude,  0°  27'  35/x.3  N. 

Longitude,  147°  45X  30x/.3. 

PROBLEM. 

(201.)   To  find  the  right  ascension  and  declination  of  a  star 

iv hen  its  latitude  and  longitude 
are  known. 

Using  the  same  figure  as  in 
the  last  problem,  and  employing 
the  same  notation,  except  that 
we  represent  EM  by  a,  we  ob- 
tain 

tang.  EM  cot.  ES=R.  cos.  SEM ; 
that  is, 

tang.  EM  =  tang,  a 
=  sin.  L  cot.  /  .  .  .  .  (A) 
Also,  ttlbF 


ECLIPTIC.  177 

Again,  sin.  PM  :  sin.  EM ::  tang.  SEM :  tang.  SPM ; 
that  is, 
sin.  (a  —  a)) :  sin.  a ::  cot.  L  :  cot.  R.  A. ::  tang.  R.  A. :  tang.  L. 


ml       .  -D    A       tang.  L  sin.  (a  —  w) 

Therefore,     tang.  R.A.=  -        — .—  —  ' 


sin.  a 

Also,  tang.  PM  cot.  PS  =  R.  cos.  SPM ; 

that  is,  tang.  Dec.  =  cot.  (a— w)  sin.  R.  A.  .  ....  (2) 

Also,          cos.  EM  :  cos.  PM  ::  cos.  ES :  cos.  PS ; 

, ,    ,  .  -p.  .       cos.  (a  —  u)  sin.  / 

that  is,  sin.  Dec.=  v (3) 

cos.  a 

And  R.  cos.  SPM=tang.  PM  cot.  PS; 

that  is,  sin.  R.  A. = tang,  (a— cj)  tang.  Dec.   .  .  (4) 

Ex.  1.  On  the  1st  of  January,  1851,  the  longitude  of  Capella 
was  79°  46X  40XX.93,  and  its  latitude  22°  51'  48/x.14  N. ;  required 
its  right  ascension  and  declination,  the  obliquity  of  the  ecliptic 
being  23°  27X  25XX.47. 

By  equation  (A),  sin.  L  =  9.9930515 

cot.  1= 0.3750341 

0  =  66°  48X  23XX.78  tang. =0.3680856 
u=23°  27/  25/x.47 
a— w=43°  20X  58XX.31 
By  equation  (1), 

tang.  L  =  0.7439461 

sin.  (a- G>)  =  9.8366072 

cosec.  a  =  0.0365991 

R.  A.  76°  25X  30".45  tang. =0.6171524 

By  equation  (2), 

cot.  (a -6>)= 0.0250345 
sin.  R.  A.  =  9.9876948 
Dec.  45°  50X  22x/.4  tang.  =  0.0127293 
By  equation  (3), 

sin.  1= 9.5894301 

cos.  (a- w)  =  9.8616417 

sec.  a= 0.4046846 

Dec. =45°  50X  22XX.4  sin. =9.8557564  . 
By  equation  (4), 

tang.  (a-G))  =  9.9749655 
tang.  Dec.  =  0.0127293 
R.  A.  =  76°  25X  30XX.4  sin.  =  9.9876948 
M 


178 


PRACTICAL    ASTRONOMY. 


In  this  example  we  have  reproduced  the  results  of  Ex.  1  in 
the  preceding  problem.  Equations  (1)  and  (2)  are  to  be  pre- 
ferred to  equations  (3)  and  (4),  for  the  reason  given  in  the  pre- 
ceding Article. 

Ex.  2.  On  the  1st  of  January,  1851,  the  longitude  of  Regu- 
lus  was  147°  45'  30x/.3,  and  its  latitude  0°  27/  35/x.3  N. ;  re- 
quired its  right  ascension  and  declination,  the  obliquity  of  the 
ecliptic  being  23°  27X  25x/.47. 

Ans.  Right  ascension,  lOh.  Om.  25.87s. 
Declination,  12°  41'  32/x.7  N. 

PROBLEM. 

(202.)  To  compute  the  longitude,  right  ascension,  and  dec- 
lination of  the  sun,  any  one  of  these  quantities,  together  with 
the  obliquity  of  the  ecliptic,  being  given. 

Let  EPQP'  represent  the  equi- 
noctial colure,  EMQ,  the  equator, 
ESQ,  the  ecliptic,  E  the  first  point 
of  Aries,  S  the  place  of  the  sun, 
PSPX  an  hour  circle  passing 
through  the  sun ;  then  EM  is  the 
sun's  right  ascension,  SM  his  dec- 
lination, ES  his  longitude,  and 
MES  the  obliquity  of  the  ecliptic. 
Then,  in  the  triangle  ESM,  we 
have 

tang.  ME  cot.  SE=R.  cos.  E  ; 
that  is,  representing  the  obliquity  by  w, 

tang.  R.  A.=tang.  Long.  cos.  G> 
_tang.  R.  A. 

cos.  w 

R.  sin.  ME  =tang.  MS  cot.  E  ; 
sin.  R.  A.  =  tang.  Dec.  cot.  w  . 
tang.  Dec.=sin.  R.A.  tang.  o>  .  .  .  . 
R.  sin.  MS  =  sin.  E  sin.  ES; 
sin.  Dec.  =  sin.  o>  sin.  Long.  . 
sin.  Dec. 
sin. 


and 

Also, 
that  is, 
and 

Also, 
that  is, 

and 


tang. 


sm. 


a) 

(2) 

(3) 
(•*) 

(5) 
(6) 


ECLIPTIC.  179 

Also,                R.  cos.  ES=cos.  ME  cos.  MS ; 
that  is,  cos.  Long.  =  cos.  R.  A.  cos.  Dec (7) 

cos.  Long. 

and  cos.  R.  A.  = 21 ....  .  /8) 

cos.  Dec. 

Ex.  1.  On  the  1st  of  June,  1852,  at  Greenwich  mean  noon, 
the  sun's  right  ascension  was  4h.  38m.  0.88s.,  and  his  declina- 
tion 22°  7X  13/x.7  N. ;  required  his  longitude. 

By  formula  (7), 

cos.  R.  A.  =  69°  30'  13". 2  =  9.5442510 

cos.  Dec.  =  9.9667958 

Longitude  =  71°  4X  20/x.3  cos.  =  9.5110468 

Ex.  2.  On  the  1st  of  January,  1852,  the  sun's  right  ascen- 
sion was  18h.  44m.  49.47s.,  and  his  declination  23°  3X  28/x.O  S. ; 
required  his  longitude. 

Ans.  280°  18'  2x/.4. 

Ex.  3.  On  the  20th  of  May,  1852,  the  sun's  longitude  was 
59°  33"  42".5,  and  the  obliquity  of  the  ecliptic  23°  21 /  29/x.06  ; 
required  his  right  ascension  and  declination. 

By  formula  (1), 

tang.  Long.  =  0.2309234 
cos.  o)  =  9.9625359 
57°  21'  32".94  tang.  =  0.1 934593 
R.  A.  =  3h.  39m.  26.20s. 

By  formula  (5), 

sin.  Long.  =  9.9355960 

sin.  w= 9.5999681 

Dec.  20°  4'  21".96  sin.  =  9.5355641 

Ex.  4.  On  the  27th  of  Octoher,  1852,  the  sun's  longitude  was 
214°  14X  45/x.2,  and  the  obliquity  of  the  ecliptic  23°  27'  30/x.69 ; 
required  his  right  ascension  and  declination. 

Ans.  Right  ascension,  14h.  7m.  56.39s. 
Decimation,  12°  56X  43/x.l  S. 


CHAPTER  VIII. 


PARALLAX. 

(203.)  THE  fixed  stars  are  so  distant  from  the  earth,  that 
their  relative  positions  are  sensibly  the  same,  from  whatever 
point  of  the  earth's  surface  we  may  view  them.  It  is  otherwise 
with  the  sun,  moon,  and  planets,  which  are  near  enough  (es- 
pecially the  moon)  to  be  displaced  by  change  of  station  on  our 
globe.  Two  spectators,  situated  on  different  points  of  the  earth's 
surface,  and  viewing  the  moon  at  the  same  instant,  do  not  see 
it  in  the  same  direction.  In  order  that  astronomers  residing  at 
different  points  of  the  earth's  surface  may  be  able  to  compare 
their  observations,  it  is  necessary  to  take  account  of  this  effect 
of  the  difference  of  their  stations,  and  it  is  convenient  to  adopt 
some  centre  of  reference  common  to  all  the  world,  to  which  each 
astronomer  may  reduce  his  observations.  The  common  point 
of  reference  universally  agreed  upon  is  the  centre  of  the  earth ; 
and  the  difference  between  the  apparent  positions  of  a  heaven- 
ly body,  as  seen  from  the  surface  or  the  centre  of  the  earth,  is 
called  its  parallax. 

PROBLEM. 

(204.)   To  find  the  parallax  of  the  moon,  etc.,  in  altitude. 

Let  C  represent  the  centre  of  the 
earth,  A  the  place  of  the  observer  on 
its  surface,  M  the  moon,  and  CAZ  the 
direction  of  a  perpendicular  to  the  sur- 
face at  A.  Then  will  the  rnoon  be 
seen  from  A  in  the  direction  AM,  and 
its  apparent  zenith  distance  will  be 
ZAM  ;  whereas,  if  seen  from  the  centre 
of  the  earth,  it  would  appear  in  the  di- 
rection CM,  with  an  angular  distance 
from  the  zenith  of  A  equal  to  ZCM ; 
so  that  ZAM  — ZCM,  or  AMC,  is  the  parallax. 


-T  PARALLAX.  181 

Let  us  put  r  =  AC,  the  radius  of  the  earth  ; 

R  =  CM,  the  moon's-  distance  from  the  earth's  cen- 

tre ; 

z  =  ZCM.,  the  moon's  true  zenith  distance; 
2T7  —  ZAM,  the  moon's  apparent  zenith  distance  ; 
#=AMC,  the  moon's  parallax  in  altitude, 
In  the  triangle  ACM,  we  have 

CM  :  CA  ::  sin.  CAM  :  sin.  AMC, 

r    . 

or  sin.  q=^  sin.  z'  ; 

K 

that  is,  the  sine  of  the  parallax  in  altitude  =  ^adius  ofearthL 

Distance  of  body 

X  sine  of  the  apparent  zenith  distance. 

The  parallax,  therefore,  for  a  given  place,  and  a  given  distance 
of  the  body  observed,  is  proportional  to  the  sine  of  its  apparent 
zenith  distance,  and  is  therefore  the  greatest  when  the  body  is 
observed  in  the  act  of  rising  or  setting,  in  which  case  its  paral- 
lax is  called  its  horizontal  parallax. 

If  we  designate  by  p  the  horizontal  parallax,  we  shall  have, 
when  ^  =  90°, 


Hence  sin.  #=rsin.  p  sin.  z'    ........  (1) 

that  is,  the  sine  of  the  parallax  in  altitude  is  equal  to  the  sine 
of  the  horizontal  parallax,  into  the  sine  of  the  apparent  zenith 
distance. 

(205.)  This  formula  furnishes  the  parallax  when  the  apparent 
zenith  distance  is  known,  but  when  the  true  zenith  distance  is 
given  we  require  a  different  formula,  which  is  obtained  as  fol- 
lows : 

The  angle  ZAM=ACM+AMC  ; 

that  is,  z'-=.z-\-q. 

Hence 
sin.  q  =  sm.  p  sin.  (z  +  q) 

=sin.  p  sin.  z  cos.  g+sin.  p  cos.  z  sin.  #,  by  Trig.,  Art.  72. 

Dividing  each  member  by  cos.  q,  we  obtain 

tang.  q  =  sin.  p  sin.  2r  +  sin.  p  cos.  z  tang.  q. 

TO-U  sm-  P  sm-  z  /«\ 

Whence  tang.  0=7-  -  .......  (2) 

1  —  sm.  p  cos.  z 


182  PRACTICAL    ASTRONOMY. 

which  formula  furnishes  the  parallax  in  altitude  when  the  true 
zenith  distance  is  known  ;  but  -the  expression  is  not  convenient 
for  computation  by  logarithms.     If  we  divide  the  numerator  of 
this  expression  by  the  denominator,  we  shall  have 
tang,  q  =  sin.  p  sin.  z  +  sin.  2p  sin.  z  cos.  z  +  sin.  3p  sin.  z  cos.  2z  +  , 

etc. 
But  by  the  Calculus,  Art.  324,  Ex.  2, 

tang.  3q 

q  =  t8LUg.  q  --  1  -  +,  etc. 
o 

Hence 
q  =  sm.  p  sin.  sr  +  sin.  2p  sin.  z  cos.  s-f-sin.  3p  sin.  z  cos.  2z 

sin.  3p  sin.  3z  .  .  v 

—  £—    —  +,  etc  ..............  (A) 

But  by  Trig.,  Art.  73, 

sin.  2z 


sin.  z  cos.  z  — 


Also,  Trig.,  Art.  79, 

sin.  3^—4  sin.  z  cos.  2z  —  sin.  z, 
and  cos.  2z—  1=—  sin.  2z. 

Therefore      sin.  z  cos.  2z  —  sin.  2r=—  sin.  3z; 
that  is, 


„ 

or  —  -  —  =     sin.  z  cos.   z  --  -  —  . 

3  o 

Therefore,  by  substitution  in  equation  (A),  we  obtain 

.  sin.  2p  sin.  2z     sin.  3p  sin.  3z 
q  =  sm.psm.z+-      ±—  —      -+-      -£—  -      -  +  ,  etc. 

If  we  wish  to  have  q  expressed  in  seconds,  we  must  divide  by 
sin.  l/x,  and  we  shall  have 


which  furnishes  the  parallax  in  terms  of  the  true  zenith  distance 
by  a  series  which  converges  rapidly. 

(206.)  The  parallax  of  the  sun  and  planets  is  so  small,  that 
we  may  employ  the  more  convenient  formula, 

q—p  sin.  z  ...........  (4) 

without  sensible  error.  But  for  the  moon,  when  the  apparent 
zenith  distance  (as  affected  by  parallax)  is  known,  we  must  make 
use  of  formula  (1)  ;  but  when  we  know  only  the  true  zenith  dis- 
tance, we  must  adopt  formula  (2)  or  (3). 


PARALLAX. 


183 


Ex.  1.  If  the  horizontal  parallax  of  Venus  is  30X/,  what  is  its 
parallax  for  an  altitude  of  30°  ? 

Solution.— By  formula  (4),  log.  30"  =  1.4771 

sin.  60°  =  9.9375 
Ans.  26/x.O  =  1.4146 

Ex.  2.  If  the  horizontal  parallax  of  Mars  is  10",  what  is  its 
parallax  for  an  altitude  of  24°  ? 

Solution.  log.  10"= 1.0000 

sin.  66°  =  9.9607 
Ans.  9".1  =  0.9607 

In  this  manner  was  computed  Table  XV.,  showing  the  par- 
allax of  the  sun  and  planets  at  different  altitudes. 

Ex.  3.  If  the  horizontal  parallax  of  the  sun  is  8x/.6,  what  is 
its  parallax  for  an  altitude  of  16°  ?  Ans.  8".27. 

Ex.  4.  If  the  moon's  horizontal  parallax  is  60'  41/x.5,  what  is 
her  parallax  when  her  apparent  zenith  distance  is  80°  19' 19"  ? 
Solution.— By  formula  (1),      sin.  60'  41".5  =  8.246833 
>,.1  sin.  80°  19'  19"    =9.993775 

Ans.  59'  49".67  sin.  =  8.240608 

Ex.  5.  If  the  moon's  horizontal  parallax  is  60'  41".5,  what 
is  her  parallax  when  her  true  zenith  distance  is  79°  19X  29V.33  ? 
Solution.'— By  formula  (3), 

sin.  6CX  41".5  =  8.246833 

sin.  79°  19'  29".33  =  9.992418 

cosec.  l/x  =  5.314425 

59'  38".29  =  3578/x.29  =  3.553676 


sin.  2j9  =  6.4937 

sin.  158°  39X= 9.5612 

cosec.  2/x  =  5.0134 


+  11".70= 1.0683 
sin.  3jf?= 4.740 
sin.  237°  58'  =  9.928?* 
cosec.  3"  =  4.837 


Hence  59X  38".29  +  ll".70-0".32 
(207.)  If  the  earth  were  a 
sphere,  a  plumb  line  suspended 
at  0  would  take  the  direction 
00,  passing  through  0,  the  cen- 
tre of  the  sphere ;  and  if  pro- 
duced upward,  it  would  meet  the 
heavens  in  Z.  This  line,  ZOO, 
would  also  be  perpendicular  to 


184 


PRACTICAL   ASTRONOMY. 


the  tangent  line  TOT'.  But 
since  the  earth  is  a  spheroid,  the 
meridian,  PEP'Q,,  is  an  ellipse, 
and  a  plumb  line  at  0  being  per- 
pendicular to  a  tangent  line,  TTX, 
takes  the  direction  of  the  normal 
line,  ON  ;  and  NO  being  pro- 
duced, meets  the  celestial  sphere 
in  Zx.  The  latitude  obtained  by  observation  will  be  expressed 
by  the  angle  Z'NQ,.  This  is  called  the  apparent  or  geograph- 
ical latitude;  while  Z  being  the  geocentric  zenith,  the  angle 
ZCQ,  is  called  the  geocentric  latitude.  The  angle  CON  is  the 
angle  which  a  vertical  line  makes  with  the  radius  of  the  earth, 
and  is  called  the  angle  of  the  vertical. 

PROBLEM. 

(208.)   To  find  the  angle  of  the  vertical. 

Let  PEP'Q,  be  a  section  of  the  earth  by  a  plane  passing 
through  the  poles.  This  section  is  an  ellipse,  whose  semi-major 
axis,  CQ,,  is  the  radius  of  the  equator,  and  whose  semi-minor 
axis,  CP,  is  half  the  polar  diameter.  If  0  be  the  position  of  an 
observer  whose  latitude  is  </>,  TTX,  a  tangent  to  the  ellipse  at  the 
point  0,  will  represent  a  horizontal  line,  and  Z'O,  which  is  per- 
pendicular to  TTX,  will  be  the  direction  of  a  plumb  line.  Rep- 
resent the  angle  OCQ,,  or  the  geocentric  latitude,  by  <//,  and 
draw  the  ordinate  OD.  Let  A  and  B  represent  the  semi-axes 
of  the  ellipse. 

B2 

By  An.  G-eom.,  Art.  80,  the  subnormal  ND=—  x9  where  x 

A. 

represents  the  abscissa  CD. 


But  OD^CD  tang.  OCD^ND  tang.  OND, 


or 


that  is, 


x  tang. 


tang. 


•  w  1 

=  -z  tang.  < 
^tang-^ 


•  B1 


The  value  of-^,  as  determined  by  Bessel,  is  0.9933254. 

A2 

Ex.  1.  Compute  the  geocentric  latitude  of  Cambridge  Ob- 
servatory, whose  geographical  latitude  is  42°  22'  48/x.6. 


.t  PARALLAX.  185 

Solution.  log.  5! = 9.99709164 

A 

tang.  42°  22'  48".6  =  9.96022854 
^5.  42°  11'  21/x.05  tang.  =  9.95732018 
Hence  the  angle  of  the  vertical  is  11'  27".55. 
Ex.  2.  Compute  the  angle  of  the  vertical  for  latitude  40°  ? 

Solution.  log.  5^= 9.99709164 

-0. 

tang.  40°  =  9.92381353 
Am.  39°  48'  40".24  tang.  =  9.92090517 
Hence  the  angle  of  the  vertical  is  11'  19".76. 
In  the  same  manner  was  computed  column  second  of  Table 
XII.,  showing  the  angle  of  the  vertical  for  every  degree  of  lati- 
tude. 


(209.)  The  horizontal  parallax  of  the  moon  is  the  angle  which 
the  earth's  radius  would  subtend  to  an  observer  at  the  moon. 
It  is,  therefore,  not  the  same  for  all  places  on  the  earth,  but 
varies  with  the  earth's  radius.  It  is  necessary,  therefore,  to 
compute  the  earth's  radius  for  the  place  of  the  observer. 

PROBLEM. 

To  compute  the  radius  of  the  earth. 

Let  EPQ,  be  half  of  the  ellipse  formed 
by  a  section  of  the  earth  through  the 
poles.  On  EQ,  describe  a  semicircle, 
and  produce  OD  to  meet  the  circum- 
ference in  I.  Join  CO  and  CI.  Represent  the  angle  OCD  by 
</>',  and  the  radius  OC  by  r.  Then,  in  the  triangle  OCD,  we 
have 

CD  =  OC  cos.  0'=r  cos.  0'  ; 
OD  =  OC  sin.  (j>'=r  sin.  0'. 

Also,  by  Conic  Sections,  Ellipse,  Prop.  12,  Cor.  3, 


But  C 

A2 
Therefore        r2  cos.  V  +  ™  •  r*  sin-  V  = 


186  PRACTICAL   ASTRONOMY. 

But,  by  Art.  208,         L» 

B2     tang,  ft7 

TT  9     /    ,       2     Sm'  V     J  A  2 

Hence         r2  cos.  2ft  +r2  •   — ZLJ  tang.  ft=A  , 

tang,  ft 

or,  multiplying  by  cos.  ft, 

r2  cos.  ft'  (cos.  ft7  cos.  ft  +  sin.  ft7  sin.  ft)=A2  cos.  ft. 
Hence,  by  Trig.,  Art.  72, 

r2  cos.  ft7  cos.  (ft7  —  ft)  =  A2  cos.  ft. 

A2  cos.  ft  .^  x 

Therefore  r2  = • —  — - (1) 

cos.  ft7  cos.  (ft7  — ft) 


.  1.  Compute  the  earth's  radius  for  Cambridge  Observato- 
ry, the  equatorial  radius  being  taken  as  unity. 

The  angle  of  the  vertical  was  found  in  the  preceding  article. 
By  formula  (1),  cos.  0  =  9.8684615 

sec. >'=:  0.1302220 

sec.  (ft7- ft)  =  0.0000024 

2)  9.9986859 

log.  r  =  9.9993429 

Ex.  2.  Compute  the  earth's  radius  for  latitude  40°. 

cos.  40°  =  9.8842540 

sec.  39°  48X  40//.24= 0.1145491 

sec.  II7  19/x.76  =  0.0000024 

2)  9.9988055 

log.  r  =  9.9994027 

PROBLEM. 

(210.)  To  find  the  horizontal  parallax  for  any  place. 

Let  P  represent  the  horizontal  parallax  for  a  place  on  the 
equator,  p  the  same  for  a  place  in  any  other  latitude ;  let  r  and 
r'  be  the  radii  of  the  earth  for  the  two  stations ;  then,  by  Art, 
204,  Rsin.  P=r; 

and,  for  the  same  reason, 

R.  sin.  p  —  r'. 

r/ 
Therefore,  sin.  p=—  sin.  P  ; 

or,  calling  the  equatorial  radius  unity, 

sin.  p^r'  sin.  P. 

As  r'  is  nearly  equal  to  unity,  we  may,  without  appreciable 
error,  adopt  the  more  convenient  formula, 


PARALLAX.  187 

j?=r".P; 

that  is,  the  moon's  horizontal  parallax  for  any  given  latitude  is 
equal  to  the  horizontal  parallax  at  the  equator,  multiplied  by 
the  radius  of  the  earth  at  the  given  latitude,  the  radius  at  the 
equator  being  considered  as  unity. 

Ex.  1.  When  the  equatorial  horizontal  parallax  is  53',  what 
is  the  horizontal  parallax  for  Cambridge  Observatory  ? 

Solution.  53' = 3180"  =  3.5024271 

r'  =  9.9993429 
Ans.  3175".19  = 3.5017700 

Ex.  2.  When  the  equatorial  horizontal  parallax  is  59',  what 
is  the  horizontal  parallax  for  latitude  40°  ? 

Solution.  59' = 3540"  =  3.5490033 

r^  9.9994027 
Ans.  3535X/.13  =  3.5484060 

It  is  this  corrected  value  of  the  equatorial  parallax  which 
should  be  employed  in  all  computations  which  involve  the  par- 
allax of  a  particular  place. 

Since  the  effect  of  parallax  is  confined  to  a  vertical  plane, 
when  the  moon  is  on  the  meridian  there  is  no  parallax  in  right 
ascension,  but  its  effect  is  wholly  on  the  declination.  In  every 
other  position  of  the  moon  (the  vertical  circle  passing  through 
the  moon  being  inclined  to  the  circle  of  right  ascension),  par- 
allax affects  the  right  ascension  as  well  as  declination  of  the 
moon. 

PROBLEM. 

(211.)   To  compute  the  parallax  in  right  ascension. 

Let  HZO  be  a  meridian, 
Z  the  geocentric  zenith  of  the 
place  of  observation,  and  P 
the  pole  of  the  equator.  Let 
A  be  the  true  place  of  the 
moon  seen  from  the  centre 
of  the  earth,  and  B  the  ap- 
parent place  seen  from  the 
surface ;  then  will  the  arc  AB  be  the  parallax  in  altitude ;  and 
the  true  hour  angle,  ZPA,  is  changed  by  parallax  into  ZPB. 
Therefore  APB  is  the  parallax  in  right  ascension. 


188  PRACTICAL   ASTRONOMY. 

Let  us  represent  the  hori- 
zontal parallax  of  the  place 
by  p ;  the  parallax  in  right 
ascension  by  II ;  the  hour  an- 
gle, ZPA  (which  is  equal  to 
the  sidereal  time,  minus  the 
moon's  true  right  ascension), 
by  h  ;  the  moon's  declination 
(which  equals  90°— AP)  by  d;  and  the  geocentric  latitude  of 
the  place  of  observation  by  $' .  The  angle  ZPB  will  then  be 
equal  to  h  +  II. 

In  the  spherical  triangle  ABP,  we  have 

.  -n     .     -D   .   .  sin.  AB  sin.  B 

sin.  AP :  sin.  B  :.  sm.  AB  :  sin.  APB  =  sm.  11= : — -r^ . 

sin.  AP 

Also,  in  the  spherical  triangle  BPZ,  we  have 

•         r>r7        •         -DlDr7  •         T>ry  13        Sm.  PZ  SHI.   (APZ  +  Il) 

sm.  BZ  :  sin.  BPZ  ::  sin.  PZ  :  sm.  B  = = — 5^ • 

sin.  BZ 

ml_      f          .  sin.  AB   sin.  PZ  sin. 

Therefore    sm.  n = .  — — -.- 

sin.  AP  sm.  BZ 

But,  by  Art.  204, 

sin.  AB=:sin.  p  sin.  BZ. 

TT  .  sin.  p  sin.  BZ  sin.  PZ  sin. 

Hence    sin.n=—          — = — r^—. — ^^ 

sin.  AP  sm.  BZ 

_sin.  p  cos.  0'  sin.  (7i-fn) 
cos.  6 

,  sin.  p  cos.  6' 

Let  us  put  a— — — — . 

cos.  6 

Then 
sin.  IL=a  sin.  (A  +  n) (1) 

—  a  sin.  h  cos.  Tl  +  a  cos.  h  sin.  n.     Trig.,  Art.  72. 
Divide  each  member  by  cos.  n,  and  we  have 

tang.  U  =  a  sin.  h  +  a  cos.  h  tang.  n. 

Therefore  tang.  n=T^^'-^T  .....  /.  .  (2) 

I  — a  cos.  ft 

This  formula  may  be  developed  in  a  series,  as  in  Art.  205,  and 
we  shall  obtain 

a  sin.  h     a2  sin.  2h  .  a3  sin.  37*  /ox 

n=-^-T7r+— — - ^77-  4— - — o77-  +  j  etc.  .  .  (3) 
sm.  1"        sin.  2"  X/ 


PARALLAX.  189 

Equation  (1)  will  furnish  the  parallax  in  right  ascension,  when 
we  know  the  apparent  hour  angle  (as  affected  by  parallax) ;  but 
when  we  know  only  the  true  hour  angle,  A,  we  must  employ 
equation  (2)  or  equation  (3). 

Ex.  1.  Find  the  moon's  parallax  in  right  ascension  for  the 
High  School  Observatory,  Philadelphia,  Lat.  39°  57/  7"  N.,  when 
the  horizontal  parallax  of  the  place  is  59'  36XX.8,  the  moon's  Dec. 
24°  5'  llxx.6  N.,  and  the  moon's  hour  angle  61°  10X  47x/.4. 
The  geocentric  latitude  of  the  place  is  39°  45X  47XX.5. 
By  formula  (3), 
sin.  p  =  8.239048 
cos.  0'  =  9.885754 
sec.  6= 0.039563  a2  =  6.3287  a3=4.493 

a = 8.164365          sin.  2h  =  9.9267          sin.  3h  =  8.79ln 
sin.  h  =  9.942572      cosec.  2"  =  5.0134     cosec.  3XX  =  4.837 
cosec.  lxx =5.314425      +  18/x.57  =  1.2688       -  0'x.01  =  S.l21n 
2638".53  =  3.421362 
Hence 

n = 2638/x.53  -f  18XX.57  -  Oxx.01 = 44X  17/x.09. 
Therefore  the  moon's  apparent  hour  angle  is 

61°  10X  47XX.4+44X  17//.l 
^61°  55'    4XX.5. 

With  this  hour  angle  the  parallax  may  be  computed  by  form- 
ula (1),  thus :  a  =  8.164365 

sin.  (h  +  n)  =  9.945604 
Ans.  sin.  44X  17/x.09  =  8.109969 

Ex.  2.  Find  the  moon's  parallax  in  right  ascension  for  the 
High  School  Observatory,  Philadelphia,  when  the  horizontal  par- 
allax of  the  place  is  57X  7XX.5,  the  moon's  Dec.  26°  23X  3XX.6  N., 
and  hour  angle  32°  39X  49XX.5. 

Solution. 

sin.  p.  =  8.220532 

cos.  (//  =  9.885754 

sec.  6= 0.047773  a2  =  6.3081  a3= 4.462 

a- 8.154059  sin.  2h  =  9.9584          sin.  3/*  =  9.996 

sin.  h  =  9.732158       cosec.  2XX  =  5.0134       cosec.  3X/  =  4.837 
cosec.  lxx  =  5.314425          19XX.05  =  1.2799  Oxx.20  = 

1587XX.24 =37200642 


190  PRACTICAL   ASTRONOMY. 

Hence  n  =  1587x/.24+19//.05  +  0//.20=26/  46XX.49. 

Therefore  the  moon's  apparent  hour  angle  is  33°  6X  36X/.0. 

By  formula  (1),  a  =  8.154059 

sin.  (h + n)  =  9.737390 
Ans.  sin.  26X  46".49= 7.891449 

Ex.  3.  Find  the  moon's  parallax  in  right  ascension  for  West- 
ern Reserve  College,  Ohio,  Lat.  41°  14'  42",  when  the  horizon- 
tal parallax  of  the  place  is  59'  36x/.5 ;  the  moon's  Dec.  24°  4X 
41XX.7  N.,  and  hour  angle  68°  9X  51/x.9.  Ans.  45X  56XX.5. 

Ex.  4.  Find  the  moon's  parallax  in  right  ascension  for  West- 
ern Reserve  College,  Ohio,  when  the  horizontal  parallax  of  the 
place  is  57X  7XX.7 ;  the  moon's  Dec.  26°  24X  31x/.5  N.,  and  hour 
angle  23°  13X  12/x.O.  Ans.  19X  12/x.6. 

In  this  manner  was  computed  Table  XVI.,  showing  the  moon's 
parallax  in  right  ascension  for  Cambridge  Observatory  for  all 
hour  angles  from  the  meridian  to  the  horizon. 

PROBLEM. 

(212.)  To  compute  the  moon's  parallax  in  declination. 

Let  Z  be  the  geocentric  ze- 
nith of  the  place  of  observa- 
tion, P  the  pole  of  the  equa- 
tor, A  the  true  place  of  the 
moon,  and  B    its    apparent 
place  ;  then  AB  is  the  paral- 
lax in  altitude,  and  BP— AP 
is  the  parallax  in  declination. 
Represent  the  true  declination,  zenith  distance,  and  hour  an- 
gle of  the  moon  by  d,  Z,  and  h  ;  and  the  apparent  values  of  the 
same  quantities  by  the  same  letters  accented,  dx,  Zx,  and  7ix. 
Also,  let  ?!  represent  the  parallax  in  declination. 
By  Trig.,  Art.  225,  we  have 

cos.  a— cos.  b  cos.  c 

COS.  A  =  —  — : —         — . 

sm.  b  sin.  c 

This  formula,  applied  successively  to  the  triangles  APZ  and 
BPZ,  gives 

cos.  AP-cos.  PZ  cos.  AZ     cos.  BP-cos.  PZ  cos.  BZ 


cos.AZP=- 


sin.  PZ  sin.  AZ  sin.  PZ  sin.  BZ 


PARALLAX.  191 

Therefore 

sin.  6—  sin.    '  cos.  Zsin.  d/  —  sin.  §'  cos.  7S 


that  is, 

sin.  6  sin.  Zx—  sin.  <//  sin.  Z7  cos.  Z=:sin.  6/  sin.  Z 

—  sin.  ($>'  sin.  Z  cos.  Z', 
or 

sin.  d  sin.  Z7  —  sin.  $'  (sin.  Z7  cos.  Z  —  sin.  Z  cos.  Z')  —  sin.  6/  sin.  Z. 
But  by  Trig.,  Art.  72,  the  factor  included  within  the  paren- 
thesis is  equal  to  sin.  (Z'  —  Z),  which,  by  Art.  204,  is  equal  to 
sin.  #,  or  sin.  p  sin.  Z'. 
Therefore 

sin.  6  sin.  Z'  —  sin.  p  sin.  $'  sin.~Z/=sin.  6/  sin.  Z, 
or  sin.  (5X  sin.  Z  =  sin.  Zx  (sin.  6—  sin.  p  sin.  (/>')  .  .  (A) 

Now,  in  order  to  eliminate  Zx  and  Z,  we  have,  in  the  spherical 
triangles  AZP  and  BZP, 

.  rro  sin.  AP  sin.  APZ  sin.  BP  sin.  BPZ 
sin.  AZP  =-  —  ;  —  -=—      -v-Tuii  --  ' 
sin.  AZ        sm.  BZ 

Therefore 

sin.  BZ  sin.  AP  sin.  APZ 


sin.  BP  sin.  AZ  =  ' 


sin.  BPE: 


r/    .      „     sm.  Z7  cos.  6  sin.  h  /TJX 

or  cos.  (5  sm.  Z— : — T—  ....  (B) 

sin.  h 

Dividing  equation  (A)  by  equation  (B),  we  obtain 

sin.  6— sin.  p  sin.  </>7     .      , 

tang.  (57  = .  —  _--  '  .sm.  7i7. 

sin.  /i  cos.  o 

TT  /sin.  c5     sin.  p  sin.  07\sin. 

Hence  tang,  d'  —  i- 

\cos.  6  cos. 


. 

1) 
sm.  6 

(213.)  Formula  (1)  furnishes  the  apparent  declination  in  terms 
of  the  true  declination,  the  true  hour  angle  and  the  apparent 
hour  angle,  which  is  obtained  by  the  preceding  problem.  It  is 
the  simplest  formula  known  for  the  parallax  in  declination  ;  but 
in  order  to  obtain  all  the  accuracy  which  is  required  in  many 
computations,  it  is  necessary  to  have  a  table  of  sines  and  tan- 
gents to  seven  decimal  places  for  every  second  of  the  quadrant. 
It  is,  therefore,  sometimes  more  convenient  to  have  a  formula 
which  shall  furnish  the  parallax  directly. 


192  PRACTICAL   ASTRONOMY. 

From  the  last  equation  but  one  we  have 

tang.  6/  sin.  h  sin.  p  sin. 

=  tang.  6 


sin.  cos. 


,  tang.  6/  sin.  7i     sin.  p  sin.  0X 

whence          tang.  6  --  =1  —  .  -  —  -  £_    -X-. 

sin.  h'  cos.  d 


,  „  tang.  d/  sin.  A     sin.  p  sin.  (i/ 

tang,  d-tang.  <J'+tang.  c5x  --  SI  —  -  =  _        _^L. 

sin.  /i  cos.  <5 


or 

tang.  6— tang,  dx+tang.  & — — 

sin. 

mn    i  $  •-•— "  fVi 

But    tang.  6 -  tang.  dx  =         l          ; .     Trig.,  Art.  76. 
cos.  6  cos.  dx 

Therefore 

sin.  (6— dx)      sin.  />  sin.  <£x     tang.  (5X  y  .      7.      .      7X 

J -  = •— — .      .     (sm.  h'  —  sm.  7i). 

cos.  d  cos.  6  cos.  d  sm.  7ix 

But  by  Trig.,  Art.  75, 

sin.  7ix  — sin.  7^  =  2  sin.  ^(h'  —  li)  cos.  i(7ix+7i) 

Therefore 

sin.  TT      _sin.  ^  sin.  </>x  2  sin.  Jn  cos.  (7i  + Jn)  tang,  d' 
cos.  d  cos.  dx           cos.  d  sin.  7ix 

But  by  Trig.,  Art.  74, 

sin.  n       ,.  ,    ,  011      sin.  p  cos.  <ix  sin.  // 

2  sin.  in= : — ,  which,  by  Art.  211  —  — 

cos.  in  cos.  d  cos.  Jn 

Therefore 

sin.  TT = sin.  p  sin.  <j>'  cos.  dx 

—  sin.  p  cos.  <£x  cos.  (7i  + Jn)  sec.  jn  sin.  dx    .   (C) 
Let  us  put 

cot.  b  =  cos.  (7i  +  Jn)  cot.  0X  sec.  Jn     ;  (D) 

Then 

sin.  7T=:sin.  p  sin.  $>'  cos.  dx— sin.  p  sin.  $'  sin.  dx  cot.  ^» 
Ycos.  dx  sin.  Z>  — sin.  dx  cos.  b\ 


—  sin.  2?  sm 


/ 

m.  0X( 
\ 


sin.  / 

sin.  (4-.T).     By  Trig.,  Art.  72. 


sm. 

sin.  #  sin. 


Let  us  also  put  c  = 

sin.  6> 

Then  sin.  rr  =  c  sin.  (6  —  (5+7i)    .......  (2) 

But  by  Trig.,  Art.  72, 

sin.  (b  —  (S-f-7r)  =  sin.  (b  —  d)  cos.  TT  +  COS.  (6  —  d)  sin.  TT. 
Therefore 


PARALLAX.  193 

sin.  77— c  sin.  (b  —  6)  cos.  TT+C  cos.  (b— 6)  sin.  TT. 
Dividing  by  cos.  rr,  we  have 

tang.  TC—C  sin.  (b  —  d)  +  c  cos.  (b— 6)  tang.  TT. 
Whence 

c  sin.  (6— 6) 


Developing  this  formula  in  a  series,  as  in  Art.  205,  we  obtain 

_c  sin.  (b-8)     c*  sin.  2(ft- J)     c3  sin.  3(6-<?) 
sin.  I'"  sin.  2"  sin.  3" 

Equation  (2)  will  furnish  the  parallax  in  declination  when  we 
know  the  apparent  declination  (as  affected  by  parallax).  But 
when  we  know  only  the  true  declination,  we  must  employ  equa- 
tion (1),  or  (3),  or  (4). 

(214.)  The  auxiliary  an- 
gle b,  introduced  in  equation 
(D),  has  a  geometrical  sig- 
nification. 

If  we  draw  the  arc  PC, 
bisecting  the  angle  APB,  we 
shall  have  APC  =  JIT,  and 
ZPC  =  h  +  Jn,  whence  equa- 
tion (D)  becomes  Q 

cot.  &  =  cos.  ZPC. tang.  PZ  sec.  APC, 
or  cot.  b  cos.  APC = tang.  PZ  cos.  ZPC    .  •.  .  .  (I) 

If  we  draw  Z6  perpendicular  to  PC,  ~Pab  will  be  an  isosceles 
triangle,  and  we  shall  have,  by  Napier's  rule, 

tang.  Pcnrtang.  Pa  cos.  APC  =  tang.  PZ  cos.  ZPC  .  (2) 

Comparing  equations  (1)  and  (2),  we  see  that  the  arc  b  is  the 
complement  of  Pa,  or  the  arc  b  is  equal  to  the  declination  of  the 
point  a.  If  we  produce  the  arcs  PA  and  PB  to  meet  the  equa- 
tor EQ,  then  aM.  or  &N  will  represent  the  arc  b. 

But  AM=;(5. 

Therefore  Aa  —  b — d. 

Also,  BN^d'^d — TT. 

Therefore  Eb  =  b  —  (d— 7r)  =  b  — 6+<rr. 

Also,  by  Spherical  Trigonometry,  Art.  215, 

sin.  ZaA  :  sin.  ZA : :  sin.  AZa :  sin.  A#, 
and  sin.  ZB  :  sin.  ZbE ::  sin.  B6 :  sin.  BZ6. 

N 


194  PRACTICAL   ASTRONOMY. 

Therefore,  since  Pab  is  an  isosceles  triangle,  we  have 
sin.  ZB  :  sin.  ZA  ::  sin.  B6  :  sin.  Ao, 
sin.  2:x_sin.  (b  —  6  -{-IT) 
sin.  z~     sin.  (b—  6) 

This  equation  will  be  employed  in  Art.  218,  page  200. 
Ex.  1.  Find  the  moon's  parallax  in  declination  for  the  High 
School  Observatory,  Philadelphia,  when  the  horizontal  parallax 
of  the  place  is  59/  36".8,  the  moon's  Dec.  24°  5X  llx/.6  N.,  the 
moon's  hour  angle  61°  10'  47/x.4,  and  the  parallax  in  right  as- 
cension 44'  17".l. 

Solution.  —  By  formula  (1),  page  191, 

sw.p=  8.2390478 

sin.  0'  =   9.8059193 

cosec.  6=  10.3892161 

.02717585  ==  8.4341832 

.97282415=  9.9880344 

sin.  h'=   9.9456035 

tang.  6=   9.6503464 

cosec.  h  =  10.0574279 

t?x  =  23°  39'  1".50  tang.=   9.6414122 

Therefore  7r=24°  5X  llxx.6-23°  39X  1^.50  =  26X  lO^.l. 

By  formula  (4),  page  193,    cos.  (h  +  in)  =  9.677980 

cot.  ^  =  0.079834 

sec.  ^n  =  0.000009 

b  =  60°  12X  22x/.2  cot.  =  9.757823 


b  -6  =36°    T  10/x.6 

sin.  p  =  8.239048 
sin.  $'  =  9.805919 

cosec.  b  =  0.061571  c2  =  6.2131  c3  =  4.320 

c  =  8.106538  sin.  2(b-  6)  =  9.9788  sin.  3(6-6)  =  9.977 

sin.  (b  -  6)  =  9.770464      cosec.  2X/  =  5.0134      cosec.  3/x  =  4.837 

cosec.  l/x  =  5.314425         16".04=  1.2053  Ox/.14=9134 

1553X/.91  =  3191427 

7r=1553x/.9l4-16//.04+0xx.14  =  26/10//.l. 
Therefore  the  moon's  apparent  declination  is 

24°    5X  llx/.6-26/  KXM 
=  23°  39X    lx/.5  N. 


PARALLAX.  195 

With  this  declination  the  parallax  may  be  computed  by  form- 
ula (2),  page  192,  thus  : 

c  =  8.106538 

sin.  (6-^)=  9.774963 

sin.  26'10".1  =  7.881501 

Ex.  2.  Find  the  moon's  parallax  in  declination  for  the  High 
School  Observatory,  Philadelphia,  when  the  horizontal  parallax 
of  the  place  is  57'  7".5,  the  moon's  Dec.  26°  23X  3XX.6  N.,  the 
moon's  hour  angle  32°  39X  49XX.5,  and  the  parallax  in  right  as- 
cension 26"  46x/.5. 

Solution.  —  By  formula  (4), 

cos.  (h  +  Jn)  =  9.924147 

cot.  <//  =  0.079834 

sec.  jn  =  0.000003 

6  =  44°  44X  13XX.7  cot.  =  0.003984 

6=26°  23'    3x/.6 


b-6=18°  2: 

sin.  p  =  8.220532 
sin.  0X  =  9.805919 
cosec.  6  =  0.152517 

Lx  10xx.l 

=  8.178968  c2=6.3579  c3  =  4.537 

sin.  (b  -6)  =  9.498128  sin.  2(b  -6)  =  9.7765  sin.  3(6  -d)  =  9.914 
cosec.  lxx  =  5.314425      cosec.  2XX  =  5.0134      cosec.  3X/  =  4.837 
980".67  =  2.991521          14X/.06  =  1.1478  0/x.19  =^288 

TT  =  980x/.67  +  14/x.06  +  0".  19  =  16X  34/x.92. 

Therefore  the  moon's  apparent  declination  is 

26°23X    3//.6-16/34//.9 
=  26°    6X  28".7N. 

"With  this  declination  the  parallax  may  be  computed  by  form- 
ula (2),  thus  : 

£  =  8.178968 

sin.  (!>-&)  =  9.504392 

sin.  16X  34x/.92  =  7.683360 

Ex.  3.  Find  the  moon's  parallax  in  declination  for  Western 
Reserve  College,  Ohio,  when  the  horizontal  parallax  of  the  place 
is  59X  36/x.5,  the  moon's  Dec.  24°  4X  41XX.7  N.,  and  hour  angle 
68°  9X  51x/.9  ;  and  the  parallax  in  right  ascension  45X  56XX.5. 

Ans.  29X  17//.9. 


196  PRACTICAL   ASTRONOMY. 

Ex.  4.  Find  the  moon's  parallax  in  decimation  for  Western 
Reserve  College,  Ohio,  when  the  horizontal  parallax  of  the  place 
is  57'  7/x.7,  the  moon's  Dec.  26°  24'  31x/.5  N. ;  the  hour  angle 
is  23°  13X  12X/.0,  and  the  parallax  in  right  ascension  19X  12x/.6. 

Ans.  16X  15/x.8. 

The  effect  of  parallax  is  always  to  increase  the  hour  angle,  or 
the  angular  distance  of  the  moon  from  the  meridian ;  hence, 
when  the  moon  is  on  the  eastern  side  of  the  meridian,  the  par- 
allax in  right  ascension  increases  the  true  right  ascension  of  the 
moon ;  hut  when  the  moon  is  on  the  western  side  of  the  merid- 
ian, the  parallax  diminishes  the  right  ascension.  The  parallax 
in  declination  increases  the  distance  of  the  moon  from  the  north 
pole  in  both  situations. 

In  the  computation  of  occultations  of  stars  by  the  moon,  it  is 
convenient  to  know  the  change  which  the  parallaxes  undergo  in 
a  given  interval  of  time,  as,  for  example,  in  one  hour.  This  may 
be  effected  by  differentiating  the  expressions  already  obtained 
for  the  parallaxes. 

PROBLEM. 

(215.)  To  find  the  hourly  variation  of  the  parallax  in  right 
ascension. 

Equation  (1),  of  Art.  211,  is 

sin.  v  cos.  &'  sin.  h' 

sin.  n= —  — . 

cos.  6 

Since  the  arcs  n  and  p  are  in  all  cases  small,  they  will  differ 
but  little  from  their  sines,  and  sin.  /ix  differs  but  little  from  sin. 
h  ;  we  will  therefore  employ  the  more  convenient  formula, 
_p  cos.  (//  sin.  h 

cos.  6 

In  this  formula  h  is  the  only  quantity  which,  by  its  rapid  va- 
riation, has  any  important  influence  on  the  quantity  sought. 
Hence,  regarding  h  as  the  only  variable,  we  obtain 
,    _p  cos.  <//  cos.  hdh 

cos.  6 

The  differential  of  h  must  be  taken  in  parts  of  radius.  If  the 
variation  is  required  for  one  hour,  dh  will  represent  the  arc  of 
15°,  which  is  .2617994,  radius  being  unity. 


PARALLAX.  197 

Ex.  1.  Find  the  hourly  variation  of  the  moon's  parallax  in 
right  ascension  for  Cambridge  Observatory,  whose  geocentric 
latitude  is  42°  ll/  21",  when  the  horizontal  parallax  of  the 
place  is  57X,  the  moon's  Dec.  25°,  and  the  hour  angle  50°. 

Solution.  jt?  =  57'  =  3420"  ==  3.534026 

cos.  0'  =  9.869778 

cos.  h  =  9.808067 

dfc  =  . 2617994  =  9.417969 

sec.  6= 0.042724 

470/x.5  =  2.672564 

Ex.  2.  Find  the  hourly  variation  of  the  moon's  parallax  in 
right  ascension  for  Cambridge  Observatory,  when  the  horizontal 
parallax  of  the  place  is  61X,  the  moon's  Dec.  20°,  and  the  hour 
angle  15°.  Am.  729x/.8. 

PROBLEM. 

(216.)  To  find  the  hourly  variation  of  the  parallax  in  dec- 
lination. 

Equation  (C),  of  Art.  213,  is 

sin.  TT  =  sin.  p  sin.  $'  cos.  6'  —  sin.  p  cos.  $'  cos.  h  sec.  ^11  sin.  &. 
Substituting  the  arcs  TT  and  p  for  their  sines,  and  using  6  in 
place  of  (5X,  we  obtain  the  following  more  convenient  formula, 
which  affords  an  approximate  value  of  TT, 

7T—p  sin.  $'  cos.  (5—  p  cos.  <j>'  cos.  h  sin.  6. 
Differentiating  this  formula,  regarding  h  as  the  only  variable, 
we  obtain 

dn=p  cos.  $'  sin.  6  sin.  hdh. 

Ex.  1.  Find  the  hourly  variation  of  the  moon's  parallax  in 
declination  for  Cambridge  Observatory,  when  the  horizontal  par- 
allax of  the  place  is  57'  the  moon's  Dec.  25°,  and  the  hour  an- 
gle 50°. 

Solution.  P  =  3.534026 

cos.  0'  =  9.869778 

sin.  (5=9.625948 

sin.  A  =  9.884254 

<ta  =  9.417969 

214".8  =  2.331975 

Ex.  2.  Find  the  hourly  variation  of  the  moon's  parallax  in 


198 


PRACTICAL   ASTRONOMY. 


declination  for  Cambridge  Observatory,  when  the  horizontal  par- 
allax of  the  place  is  61',  the  moon's  Dec.  20°,  and  the  hour  an- 
gle 15°.  Ans.  62".8. 


(217.)  The  apparent  diameter  of  the  moon  is  the  angle  which 
its  disk  subtends.  This  angle  is  not  the  same  for  all  points  of 
the  earth,  on  account  of  their  different  distances  from  the  moon. 
As  the  moon  rises  above  the  horizon  (if  we  suppose  its  distance 
from  the  centre  of  the  earth  to  remain  constant),  its  distance 
from  the  place  of  observation  must  dimmish  while  its  altitude 
increases,  and,  consequently,  its  apparent  diameter  .must  increase. 

PROBLEM. 

To  find  the  augmentation  of  the  moon's  semi-diameter  on 
account  of  its  altitude  above  the  horizon. 

Let  C  and  M  be  the  cen- 
tres of  the  earth  and  moon, 
and  A  a  point  on  the  earth's 
surface.  The  semi-diameter 
of  the  moon,  as  seen  from  C, 
is  the  angle  BCM  ;  but  the 
semi-diameter,  as  seen  from 
A,  is  the  angle  B'AM. 

Represent  the  angle  BCM 
by  S  ;  the  angle  Bx  AM  by  &'  ; 
the  angle  ZCM  by  Z,  and  the 
angle  ZAM  by  Z'. 
Then,  in  the  right-angled  triangle  BCM,  we  have 


sin.  BCM  =  sin.  S  = 

CM 


Also,  in  the  triangle  B'AM,  we  have 


sin.  B'AM-^sin.  S'  = 


BXM 
AM' 


Hence    sin.  S  :  sin.  S' 


BM    B'M 


::AM:CM. 


CM    AM 
But  in  the  triangle  CAM  we  have 

AM  :  CM  ::  sin.  ACM  :  sin.  CAM  ::  sin.  Z  :  sin.  Z'. 
Therefore        sin.  S:sin.  /Seisin.  Z  :  sin.  Zx, 


PARALLAX.  199 

~,     sin.  S  sin.  7/ 

or  sm.  S/= : — = (A) 

sm.  Z 

But  since  S  never  amounts  to  17X,  we  may  substitute  the  arc 
for  its  sine,  and  we  obtain 

S .  sin.  Zx 


Hence  S'-S=S. 


sin.  Z 
sin.  Zx— -sin.  Z 


sin.  Z 

which  represents  the  augmentation  of  the  moon's  semi-diame- 
ter, as  seen  from  a  point  on  the  earth's  surface  instead  of  its 
centre  ;  Z  being  the  zenith  distance  of  the  moon  viewed  from 
the  centre,  and  Zx  the  zenith  distance  as  seen  from  the  surface. 
But  by  Trig.,  Art.  75, 

sin.  Zx—  sin.  Z  =  2  sin.  |(ZX  —  Z)  cos.  J(Z'-f-Z). 
Hence 

.sm.  J(Z'-Z)  cos. 


sin. 
If  we  represent  the  parallax  in  altitude  by  #,  we  shall  have 


Hence         x  =  ^jjE  -  sin.  }  q  cos.  (Zx  -  $q). 
sin*  (^  —  w) 

But  since  the  angle  q  is  always  small,  we  may,  without  sensi- 
ble error,  put  q  for  sin.  q,  and  make  cos.  q  equal  to  unity.    Hence 
^S.gcos.  (Z'-jg) 

sin.  (Zx-g) 
But  by  Trig.,  Art.  72, 

cos.  (Zx  —  £q)  =  cos.  Zx  cos.  jg+sin.  Zx  sin.  J#. 
Also,       sin.  (Zx  —  g)  =  sin.  Zx  cos.  q—  cos.  Zx  sin.  q. 

S.qtcos.  Z'  +  ^q  sin.  Zx) 
Hence  x  =  -  ^  —  —  -         -—  -  '-. 

sm.  Zx  —  q  cos.  Zx 

But,  according  to  Burckhardt's  Tables  of  the  Moon,  we  have 

S:p::  1:3.6697. 

If  we  represent  3.6697  by  &,  then 
p  =  k.S. 
And  by  Art.  204, 

q  =  k.Ssin.  Zx. 
Therefore 

_k.S2  sin.  7/  (cos.  Zx+p.£  sin.  2ZX) 
~~l^Z^k7S  sin.  Zx  cos.  Zx     ~' 


200  PRACTICAL   ASTRONOMY. 


_k.S2(cos.  Z'  +  tk.S  sin.  2ZQ 

I-k.Scos.  Z' 

Dividing  the  numerator  by  the  denominator  in  order  to  de- 
velop this  expression  into  a  series,  we  obtain 

x  =  k.S2  cos.  Zx  +  JA2;S3  +  JAr'.S3  cos.  2Z'  +  ,  etc. 
If  we  put  k  =  /c  sin.  l/x^  0.00001779,  we  shall  have  for  the 
augmentation  expressed  in  seconds, 

x=AS2  cos.  Zx  +  iA2;S3  +  JA2S3  cos.  2ZX  +  ,  etc.  .  (1) 
By  this  formula  was  computed  Table  XIII.,  by  which  the 
augmentation  of  the  moon's  semi-diameter  may  be  obtained  by 
inspection. 

(218.)  When  the  parallax  in  declination  has  been  previously 
computed,  the  following  method  is  preferable  : 
By  Art.  214,  page  194, 

sin.  Zx__sin.  (b  —  6-\-  IT) 
sin.  Z        sin.  (b  —  6) 
Hence,  from  equation  (A),  page  199, 

^/_sin.  S  sin.  (b  —  S-\-TT) 
I  ;  sin.  (b-6)~ 

,       .      0/       .      ~     sin.  ${sin.  (b  —  d-h-rr)  —  sin.  (b  —  6)} 
and     sm.  S'  —  sin.  S  =  —  —  ^  -  ". 

Sill.    (0  —  0) 

But  by  Trig.,  Art.  75, 

sin.  (b  —  d  +  7r)  —  sin.  (b  —  6)  =  2  sin.  JTT  cos.  (b  — 
Therefore 


c         _  .         -  _  * 

sm.  (b  —  6) 

But  since  the  arcs  S,  S',  and  IT  are  very  small,  we  may  put 
$  =  sin.  /S,  and  2  sin.  ^7r  =  sm.  TT. 

TT  o/o      $  sin.  TT  cos.  (b  —  6+  JTT) 

Hence       x  ~  S'  —  S  =  -  ,  —  -±  —  -  —  -1—  - 

sm.  (b  —  6) 

But  by  Trig.,  Art.  72, 

cos.  (b  —  <S+i7r)  —  cos.  (b  —  d)  cos.  ^TT  —  sin.  (b  —  6)  sin.  |TT. 
Hence 

_  S  .  sin.  TT  {cos.  (6  —  (5)  cos.  JTT—  sin.  (6  —  (5)  sin.  JTT} 

sin.  (ft-d) 
or  x=  S  sin.  TT  cot.  (6—  6)  cos.  ITT—  S  sin.  TT  sin.  J?r. 

If  we  assume  cos.  JTT  equal  to  unity,  and  sin.  ^TT  equal  to 
sin.  TT,  we  shall  have 

rc=>S  sin.  TT  cot.  (b  —  6)  —  £S  sin.  2rr  ....  (2) 


PARALLAX.  201 

When  we  know  the  moon's  apparent  altitude,  we  may  com- 
pute its  apparent  diameter  by  equation  (1),  or  take  it  directly 
from  Table  XIII. ;  but  when  the  parallax  in  decimation  has  been 
computed,  it  is  better  to  employ  equation  (2.)  The  value  of 
(b  -6)  is  obtained  by  Art.  213,  page  192. 

Ex.  1.  Calculate  the  augmentation  of  the  moon's  semi-diam- 
eter when  its  true  semi-diameter  is  16'  30/x,  and  its  apparent 
altitude  66°. 

Solution. — By  equation  (1),  page  200, 

16"  30"  =  990"  =  2.99564  S3  =  8.9869 

2  A2  =  0.5004 

S2 = 5.99128  0.5 = 9.6990 

A=5.25021  0".15  =  9.1863 

cos.  Z7  =  9.96073  cos.  2ZX  =  9.9215 

15".93  =  1.20222  0/x.13  =  9.1078 

Hence  15//.93  +  0//.15  +  0//.13  =  16".21,  the  augmentation, 
the  same  as  given  in  Table  XIII. 

Ex.  2.  Calculate  the  augmentation  of  the  moon's  semi-diam- 
eter in  Ex.  1,  Art.  214,  when  the  horizontal  semi-diameter  is 
16'  16".0. 

Solution. — By  equation  (2),  page  200, 

16X  16"  =  976"  =  2.98945  488"  =  2.688 

77=26'  10".l     sin.  =  7.88150          sin.  27r=5.763 

b-6=36°  T  11"  cot.  =  0.13683  0".03  =  a451 

10'/.18  =  1.00778 
Hence  the  augmentation  =10//.18  — 0//.03  =  10//.15. 

Ex.  3.  Calculate  the  augmentation  of  the  moon's  semi-diam- 
eter in  Ex.  2,  Art.  214,  when  the  horizontal  semi-diameter  is 
157  37".l.  Ans.  13".6. 

When  the  moon's  hour  angle  is  known,  its  altitude  may  be 
taken  from  a  celestial  globe  with  sufficient  precision  to  furnish 
the  augmentation  of  its  semi-diameter  within  one  or  two  tenths 
of  a  second,  by  means  of  Table  XIII. 

Ex.  4.  Calculate  the  augmentation  of  the  moon's  semi-diam- 
eter in  Ex.  3,  Art.  214,  when  the  horizontal  semi-diameter  is 
16' 16".0.  Ans.  8/x.84. 


CHAPTER  IX. 

MISCELLANEOUS  PROBLEMS. 
INTERPOLATION    BY  DIFFERENCES. 

(219.)  IT  is  frequently  required,  from  a  series  of  equidistant 
terms  following  any  law  whatever,  to  deduce  some  intermedi- 
ate term.  Thus,  in  the  Nautical  Almanac,  we  have  given  the 
moon's  right  ascension  for  every  hour  of  the  day,  and  from  these 
data  it  may  be  required  to  determine  its  right  ascension  for  some 
intermediate  instant.  This  is  effected  by  interpolation.  The 
quantities  upon  which  the  values  of  the  given  magnitudes  de- 
pend are  called  Arguments.  Time  is  generally  the  argument 
in  astronomical  tables. 

Let  a,,,  a,,  a,  aQ  of  of'  of" 

be  the  given  places  of  the  moon,  corresponding  to  the  times 

T_3A,  T-2/i,  T-A,  T,  T  +  /*,  T  +  2/*,  T  +  3A, 
where  h  may  represent  any  interval  of  time  at  pleasure.  These 
places  may  be  right  ascensions  or  declinations,  longitudes  or 
latitudes,  or  magnitudes  of  any  other  kind.  Subtract  the  first 
term  of  the  series  from  the  second,  the  second  from  the  third, 
and  so  on,  giving  to  each  remainder  the  sign  which  results  from 
the  rules  of  algebra  ;  and  let  the  first  order  of  differences  be  rep- 
resented by  b,,,  b,,  b,,  etc.  Subtract  each  of  these  first  differ- 
ences from  the  one  next  below  it,  for  a  second  order  of  differ- 
ences, paying  attention  to  the  signs,  and  represent  these  differ- 
ences by  c///  c,,  c,,  etc.,  and  proceed  in  the  same  manner  for  the 
third,  fourth,  etc.,  orders  of  differences,  as  represented  in  the  fol- 
lowing table : 


MISCELLANEOUS    PROBLEMS. 


203 


Time  or 
Argument. 

Quantities. 

1st  Diff. 

2d  Diff. 

3d  Diff. 

4th  Diff. 

5th  Diff. 

6th  Diff. 

T-3A 

«/// 

T-2/i 

a 

/// 

6>/// 

&// 

d/y/ 

T-  h 

a, 

C/y 

e/// 

b/ 

£ 

//// 

T 

fto 

*/ 

e/x 

g*/// 

7 

,; 

f    _ 

o 

?V/ 

T+  h 

a' 

Co 

e/ 

^// 

6/ 

d0 

// 

T-j-2^ 

ttx/ 

Cx 

g0 

z>/x 

d' 

T  +  3A 

a'" 

C/x 

T  +  47* 

a"" 

t 

If  we  put  «(t)  to  represent  that  term  of  the  series  which  fol- 
lows a0  at  the  interval  £,  then,  as  shown  in  Algebra,  Art.  297, 
we  shall  have 

.do+,etc.  .  (A) 


Ex.  1.  Given  the  moon's  right  ascension  as  follows  : 


Date. 

Right  Ascension. 

1st  Difference.       |     2d  Difference. 

1855.                  h. 

h,     m.          s. 

m.          ,». 

5. 

February  1,  0 

8  34  36.65 

+  2     565 

1 

8  36  42.30 

-0.21 

+  2     5.44 

2 

8  38  47.74 

-0.21 

+  2     5.23 

3 

8  40  52.97 

-023 

+  2     5.00 

4 

8  42  57.97 

Required  the  moon's  right  ascension  at  February  1,  Oh.  15m. 
Here  a0  =  8h.  34m.  36.65s.  ;  b0  =  +  2m.  5.65s.  ;  c0  =  -  0.21s.  ; 
and  t  —  15m.  =  .25  in  parts  of  an  hour.     Therefore 

0  21 
a(t)  =  8h.  34m.  36.65s.  +  125.65  x  .25  +      =  x  .25  x  .75 


=  8h.  34m.  36.65s.  +  31.41s. 
=  8h.  35m.    8.08s. 
Table  XXIII.,  page  393,  gives  thft  onafficiflnts  of  each  order 


204 


PRACTICAL   ASTRONOMY. 


of  differences  for  every  hundredth  part  of  the  unit  of  time  elaps- 
ing between  the  given  terms  of  the  series.  In  the  preceding 
example  the  second  differences  are  sensibly  constant.  In  the 
following  example  the  numbers  appear  more  irregular. 

Ex.  2.  Given  the  right  ascension  of  the  moon's  limb  for  the 
upper  and  lower  transit  at  Washington,  as  follows  : 


Date. 

Right  Ascens. 

1st  Diff. 

2d  Diff. 

3d  Diff 

4th  Diff. 

5th  Diff. 

1855. 

fi.    m.       s. 

m.       s. 

s. 

s. 

s. 

s. 

July  3,  L.  T. 

22  37  59.54 

+  27  57.01 

U.T. 

23     5  56.55 

-52.95 

+27    4.06 

+  1027 

4,  L.  T. 

23  33     0.61 

-42.68 

+0.64 

+  26  21.38 

+  10.91 

-0.63 

U.T. 

23  59  21.99 

-31.77 

+  0.01 

+  25  49.61 

+  10.92 

i   -0.40 

5,  L.  T. 

0  25  11.60 

-20.85 

-0.39 

+  25  28.76 

+  10.53 

!    -0.17 

U.T. 

0  50  40.36 

-10.32 

—0.56 

+  25  18.44 

+   9.97 

6,  L.  T. 

1   15  58.80 

-  0.35 

+25  18.09 

U.T. 

1  41   1689 

to  find  the  moon's  right  ascension,  July  3,  at  its  transit  over  a 
place  one  hour  west  of  Washington. 

1677.01s.    52.95s.  x  11    10.27s.  x  11x23 
a«=23L  37m.  59.54,+-  -— 


0.64s.  x  11  x  23  x  35        0.63s.  x  11  x  23  x  35  x  47 


24x12x12x12x12     120x12x12x12x12x12' 


or 


a(t)  =  22h.  37 m.  59.54s.  +  139.75s. +  2.06 +  0.25 -0.01 -0.01 
=  22h.  40m.  21.58s. 

(220.)  It  will  generally  be  found  more  convenient  in  practice 
to  take  the  coefficients  for  the  several  orders  of  differences  direct- 
ly from  Table  XXIII.  It  will  be  observed  that  the  coefficients 
of  the  second  and  fourth  differences  are  negative,  while  those  of 
the  odd  differences  are  positive.  Hence  the  corrections  for  the 
odd  differences  will  have  the  same  sign  as  those  differences ;  but 
the  corrections  for  the  even  differences  will  have  a  sign  contrary 
to  those  differences.  Hence,  in  the  above  example,  the  correc- 
tions for  the  first,  second,  and  third  differences  are  positive,  while 
the  other  two  corrections  are  negative. 

(221.)  Formula  (A)  proceeds  from  values  which  belong  to  a 
loss  argument,  to  those  whioh  belong  to  a  greater  argument  J 


MISCELLANEOUS   PROBLEMS.  205 

but  we  are  at  liberty  to  proceed  in  the  reverse  order.  Conceive 
the  times  and  quantities  given  on  page  203  to  be  written  in  an 
inverted  order,  so  that  the  table  shall  begin  with  the  last  value, 
a///x,  and  end  with  the  first,  a,,,.  The  first  differences  would 
then  be  a'"  -  a""  =  -  b'"  ;  a"  -  a,'"  =  -  b",  etc.  That  is,  the 
first  differences  would  be  the  same  as  given  in  the  preceding  ta- 
ble, but  with  contrary  signs.  The  second  differences  would  be 
-b"-(-  b'"}  =  b'"  -b"=  +  c"  ;  that  is,  the  second  differences 
would  retain  the  same  signs  as  before.  The  third  differences, 
c/  —  cf/  —  —d',  etc.,  change  their  signs,  while  the  fourth  differen- 
ces remain  unchanged,  and  so  on  ;  that  is,  the  differences  of  an 
odd  order  have  their  signs  reversed. 

Suppose  now  that  t  is  a  proper  fraction,  representing  the  dis- 
tance of  the  term  &(t)  from  a0  ;  then  the  first,  second,  third, 
fourth,  etc.,  differences  corresponding  will  be  —b0;  +  c,  ;  —  d,,  ; 
4-  e/x/,  etc.  ;  and  consequently, 


or 


(222.)  Equations  (A)  and  (B)  are  each  of  them  only  approxi- 
mate ;  but  when  the  error  of  one  is  positive,  the  error  of  the 
other  will  generally  be  negative,  so  that  we  shall  obtain  a  more 
accurate  expression  if  we  take  the  half  sum  of  both  (A)  and  (B), 
and  we  shall  have 


_L/    ,  0 

'°°~~ 


t(t-l)  ( (t- 


But 
and  consequently, 


b0_ 

~~ 


Also,  dQ  =  d,  +  e,  ;  d/t  —d,—  e,,,  etc. 


206  PRACTICAL   ASTRONOMY. 

Substituting  these  values  in  the  preceding  equation,  we  ob- 
tain 


Since  t  is  supposed  to  be  included  between  0  and  1,  it  is  plain 
that  the  coefficients  of  the  third,  fourth,  etc.,  differences  in  form- 
ula (C)  are  smaller  than  in  formulas  (A)  and  (B)  ;  that  is,  this 
series  converges  most  rapidly. 

It  will  readily  be  perceived  that  in  formula  (A)  the  first,  sec- 
ond, third,  etc.,  differences  in  the  table  on  page  203  lie  in  a  di- 
agonal, which  starts  from  between  a0  and  a'  and  inclines  down- 
ward. In  formula  (B),  on  the  contrary,  they  lie  in  a  diagonal 
which  inclines  upward  ;  while  in  formula  (C)  the  odd  differences 
are  intersected  by  a  horizontal  line,  which  starts  from  between 
a0  and  a'  ;  but  for  the  even  differences  we  employ  the  half  sum 
of  that  which  lies  above  and  that  which  is  below  the  horizontal 
line. 

(223.)  If  we  represent  the  coefficients  of  the  several  orders  of 
differences  by  B,  C,  D,  etc.,  formula  (C)  may  be  written 

a^^av  +  Bb  +  Cc+Dd+Ee+Ff,  etc.  .  .  .  (D) 
where  we  put  b  =  b0 


d=d, 


2 

f=f,, 


2.3 


2.3.4 

--j) 


2.3.4.5 
This  formula  is  the  one  recommended  by  Professor  Bessel. 


MISCELLANEOUS    PROBLEMS. 


207 


Table  XXIII. ,  page  392,  gives  the  values  of  the  preceding  coef- 
ficients for  every  hundredth  part  of  the  unit  of  time.  It  will  be 
observed  that  the  coefficient  of  the  second  differences  is  invaria- 
bly negative  ;  but  for  values  of  t  less  than  .50,  the  coefficients  of 
the  third  and  fourth  differences  are  positive,  and  the  fifth  nega- 
tive ;  while  for  values  of  t  greater  than  .50,  the  coefficients  of  the 
third  differences  are  negative,  but  the  fourth  and  fifth  are  positive. 

Example.  Required  the  moon's  right  ascension  for  July  10, 
1855,  at  8h.  mean  time. 

Take  from  the  Almanac  three  places  of  the  moon  preceding 
and  three  places  following  the  proposed  time,  and  find  their  dif- 
ferences as  in  the  following  table  : 


Date. 

R  A. 

1st  Diff. 

2d  Diff. 

3d  Diff. 

4th  Diff. 

5th  Diff. 

A. 

h.  m.       s. 

m.        s. 

s. 

s. 

s. 

s. 

July  9,    0 

3  20  56.61 

+  26     5.41 

12 

3  47     2.02 

+27.57 

+26  32.98 

-1.89 

"  10,    0 

4  13  35.00 

+  25.68 

-2.19 

+26  58.66 

-4.08 

-0.06 

12 

4  40  33.66 

+21.60 

-2.25 

+27  20.26 

-6.33 

"  11,    0 

5     7  53.92 

+  15.27 

+  27  35.53 

12      5  35  29.45 

For  July  10,  at  Oh.,  which  is  the  date  next  preceding  the  one 
proposed,  we  find 

ac  =  4h.  13m.  35.00s.;  b  =  26m.  58.66s. 
c0  =  21.60s.;  ^=25.688.;  vc  =  J(c0  +  e/)=23.64s. 
d=  -  4.08s.  ;/=-  0.06s. 

e/=  -2.25s.;  «„=- 2.19s.;  ••.•e  =  l(e,+e//)=  -2.22s. 
The  difference  between  the  proposed  time  and  July  10  at  Oh. 
is  8h.,  which  is  two  thirds  of  the  interval  between  the  dates  in 
the  table.     Therefore  we  have 

Eb=     f  (26m.  58.66s.)    =    +  17m.  59.107s. 
Cc=:  _i(23.64s.)  -2.627s. 

Dd=-.  00617  x-  4.08s.  =  +   .025s. 

Ee  =  +. 02057  x  -2.22s.  =  -   .046s. 

F/=  +  .00069  x  -  0.06s.  = .OOOs. 

=    +  17m.  56.46s. 
«0=4h.  13m.  35.00s. 


Moon's  R.  A.  at  8h.  =  4h.  31m.  31.46s. 


208  PRACTICAL   ASTRONOMY. 

(224.)  Most  of  the  numbers  in  the  Nautical  Almanac  are  com- 
puted for  intervals  of  either  12  or  24  hours.  The  right  ascen- 
sion of  the  moon's  bright  limb  is  given  for  both  the  upper  and 
lower  culminations,  .that  is,  for  intervals  of  12  hours  of  longi- 
tude. If  we  wish  to  interpolate  for  any  other  meridian,  we  must 
consider  12  hours  as  the  unit  of  time,  and  it  is  desirable  to  have 
the  coefficients  computed  for  convenient  fractions  of  12  hours. 
When  the  computation  is  performed  by  logarithms,  it  is  con- 
venient to  have  the  logarithmic  coefficients  arranged  in  a  table. 
This  has  accordingly  been  done  in  Table  XXIV.,  which  furnish- 
es the  logarithms  of  Bessel's  coefficients  for  every  five  minutes 
throughout  12  hours.  If  the  numbers  between  which  we  wish 
to  interpolate  are  given  for  intervals  of  24  hours,  as  the  sun's 
places  in  the  Nautical  Almanac,  we  may  avail  ourselves  of  the 
same  table  of  coefficients  by  simply  doubling  each  of  the  num- 
bers in  column  first.  Thus,  when  the  interval  is  12  hours,  the 
coefficients  for  an  argument  of  one  hour  will  be  the  same  as  for 
an  argument  of  two  hours  when  the  interval  is  24  hours. 

The  preceding  example  is  most  conveniently  solved  by  the  use 
of  these  coefficients.  Taking  the  logarithms  of  the  coefficients 
B,  C,  D,  E,  and  F  from  the  table,  and  the  logarithms  of  b,  c,  d, 
e,  and  /  as  given  on  the  preceding  page,  we  have 

log.  B  =  9.8239087;  log.  C  =  9.04576w;  log.  D  =  7.79048ra; 
log.  6  =  3.2091556  log.  c=  1.37365  log.  ^= 


3.0330643  0.41941?*  840114" 

Nat.  num.  +  1079.107s.  -  2.627s.  +  0.025s. 

log.  E  =8.31336  ;  log.  F  =  6.83624 
log.  e=  0.34635^   log.  /=  8.77815^ 
8.65971w  5.61439/z 

Nat.  num.-  0.046s.  .000 

Adding  to  log.  B,  log.  C,  etc.,  the  factors  log.  £,  log.  c,  etc., 
we  obtain  for  the  several  corrections 

B&  -  =  +17m.  59.107s. 
Cc    =  -2.627s. 

Dd    =  +  0.025s. 

Ee    =  -0.046s. 

F/    =  O.OOOs. 

Sum  =+  17m.  56.46s. 
the  same  as  found  on  page  207. 


MISCELLANEOUS   PROBLEMS.  209 

If  we  neglect  the  third  and  following  differences,  that  is,  if  wo 
suppose  the  second  differences  constant,  formula  C  becomes 


"We  accordingly  take  from  the  Nautical  Almanac  four  consec- 
utive arcs,  such  that  the  arc  sought  may  fall  between  the  two 
middle  ones,  and  for  the  second  difference  we  employ  the  mean 
of  the  two  second  differences  thus  obtained. 

In  the  example  on  page  207,  if  we  regard  only  first  and  sec- 
ond differences,  we  shall  obtain  the  moon's  right  ascension, 

4h.  31m.  31.48s. 
instead  of  4h.  31m.  31.46s. 

The  error  arising  from  neglecting  the  third  and  following  dif- 
ferences amounts,  therefore,  only  to  0.02s. 

PROBLEM. 

(225.)  To  find  the  time  of  conjunction  or  opposition  of  the 
moon  with  the  sun. 

The  right  ascension  of  the  moon  is  given  in  the  Nautical  Al- 
manac for  every  hour  of  the  day,  and  the  right  ascension  of  the 
sun  is  given  for  noon  of  each  day.  An  inspection  of  these  col- 
umns will  readily  show  between  what  hours  conjunction  or  op- 
position takes  place.  Take  out  four  successive  right  ascensions 
of  the  moon,  such  that  the  phase  sought  shall  fall  between  the 
second  and  third  of  the  hours,  and  find  by  interpolation  the  cor- 
responding right  ascension  of  the  sun.  For  each  hour  subtract 
the  right  ascension  of  the  sun  from  that  of  the  moon  ;  the  dif- 
ferences will  represent  the  distances  of  the  moon  from  the  sun. 
Then,  if  only  an  approximate  result  is  desired,  we  may  determ- 
ine by  a  simple  proportion  when  the  difference  of  right  ascension 
amounts  to  zero,  or  twelve  hours.  But  if  a  more  accurate  result 
is  desired,  we  must  take  account  of  the  second  differences. 

Example. 

Required,  the  "Washington  mean  time  of  opposition  in  right 
ascension  of  the  sun  and  moon,  October  24,  1855. 

We  readily  discover  by  an  inspection  of  the  ephemeris  that 
opposition  takes  place  between  19h.  and  20h.,  Greenwich  time. 
We  then  take  from  the  Nautical  Almanac  the  right  ascension  of 

0 


210 


PRACTICAL    ASTRONOMY. 


the  sun  and  moon  for  four  successive  hours,  and  find  their  differ- 
ences, neglecting  the  12  hours,  as  follows : 


Date,  j         Sun's  R.  A.         |      Moon's  R.  A. 

Moon  —  Sun. 

1st  Diff. 

2d  Diff. 

A.      1    h.      m.           s. 

h.      m.         s. 

m.        s. 

m.       s. 

A1. 

18 

13  56  30.45 

1  53  48.67 

-2  41.78 

+  2  3.78 

19 

13  56  40.01 

1  56     2.01 

-     38.00 

+  0.16 

+  2  3.94 

20 

13  56  49.57 

1  58  15.51 

+  1  25.94 

+  0.15 

+  2  4.09 

21 

13  56  59.13 

2     0  29.16 

+  3  30.03 

If  we  neglect  the  second  differences,  the  time  of  opposition 
may  be  found  by  the  proportion 

2m.  3.94s. :  3600s. ::  38.00s. :  18m.  23.8s. ; 
that  is,  opposition  takes  place  at  19h.  18m.  23.8s.,  and  this  re- 
sult is  within  half  a  second  of  the  truth.     If  greater  accuracy  is 
required,  we  must  take  account  of  the  second  differences,  which 
may  be  done  as  follows  : 

In  the  formula  of  interpolation,  page  206, 

,  etc., 

t  must  be  regarded  as  the  unknown  quantity,  all  the  others  being 
known. 

Developing  this  formula,  we  have 

(t)_          _!_/     b—^— 

~2     ~2' 
c 


or 


Now  the  approximate  value  of  t  is 
this  value  above,  we  obtain 


Substituting 


r       r    //(t)      n    ' 
h—     -\-  ~     ° 

U  '"1^       •  - 


which  is  a  more  accurate  value  of  t.  In  the  present  case1  a(t)  be- 
comes zero,  because  we  wish  to  determine  when  the  difference 
of  right  ascension  between  the  sun  and  moon  is  zero  (neglecting 
the  12  hours)  ;  hence  we  have 


-a 


,      c     c  a0} 
o—~-—  —  .  — 


MISCELLANEOUS   PROBLEMS. 


211 


where  we  must  be  careful  to  preserve  the  proper  sign  of  each 
term. 

Substituting  the  values  of  these  letters,  we  have 

38.00  _  38.00 

~  123.94  -  .08  +  .03  ~~  123.89' 
where  t  is  expressed  in  parts  of  an  hour. 

Multiplying  by  3600  to  reduce  the  result  to  seconds,  we  ob- 
tain 1104.2s.,  or  18m.  24.2s. ;  whence  the  corrected  time  of  op- 
position is 

19h.  18m.  24.2s.  Greenwich  mean  time, 
or  14h.  10m.  13.0s.  Washington  mean  time. 

PROBLEM. 

(226.)  To  find  the  hourly  motion  of  the  moon  in  right  as- 
cension^ etc. 

The  moon's  hourly  motion  may  be  found  very  nearly  by  tak- 
ing the  difference  between  two  successive  numbers  in  the  Nau- 
tical Almanac,  the  one  before  and  the  other  after  the  time  for 
which  the  hourly  motion  is  wanted.  If  the  proposed  instant 
does  not  fall  midway  between  the  two  dates  in  the  Almanac,  we 
must  apply  a  correction  by  taking  a  proportional  part  of  the  sec- 
ond difference. 

Example  1. 

Required  the  moon's  hourly  motion  in  right  ascension,  Octo- 
ber 24,  1855,  at  19h.  18m.  24.2s.,  Greenwich  mean  time. 
We  take  from  the  Nautical  Almanac  the  following  numbers ; 


Date. 

Moon's  R.  A. 

1st  Diff. 

2d  Diff. 

h. 

18 
19 
20 

h.    m.          s 

1  53  48.67 
1  56     2.01 
1  58  15.51 

m.        s. 

+  2  13.34 
+  2  13.50 

f. 

H-0.16 

The  change  of  the  moon's  right  ascension  from  18h.  to  19h. 
is  2m.  13.34s.,  which  may  be  regarded  as  the  hourly  motion  for 
18h.  30m.  In  the  same  manner,  the  hourly  motion  for  19h.  30m. 
is  2m.  13.50s.  In  order  to  obtain  the  hourly  motion  for  19h. 
18m.  24.2s.,  we  state  the  proportion 

60m. :  0.16s. ::  48m.  24.2s. :  0.13s., 


212  PRACTICAL   ASTRONOMY. 

which,  being  added  to  2m.  13.34s.,  gives  the  hourly  motion  for 
19h.  18m.  24.2s.,  equal  to  2m.  13.47s.  in  time,  or  33'  22" .05 
in  arc. 

In  calculating  eclipses,  it  is  necessary  to  know  the  hourly  mo- 
tion of  the  moon  from  the  sun.  This  is  obtained  by  finding  the 
sun's  hourly  motion  in  the  manner  already  explained,  and  sub- 
tracting it  from  the  moon's  hourly  motion.  Thus,  if  the  sun's 
hourly  motion,  October  24,  was  9.56s.,  then  the  hourly  motion 
of  the  moon  from  the  sun  was  2m.  13.47s.  —  9.56s. = 2m.  3.91s. 
in  time,  or  30'  58/x.65  in  arc. 

(227.)  The  preceding  method  is  slightly  inaccurate  in  princi- 
ple, and  when  the  interval  between  the  moon's  places  amounts 
to  12  hours,  the  error  can  not  be  neglected.  An  accurate  form- 
ula for  the  hourly  motion  may  be  obtained  by  differentiating 
equation  D,  page  206.  We  thus  find 

dKl_A.2<-l.  **-»+*      4^-6^-2^+2 
~W          ~~        "0~~  "27374" 

If  the  moon's  places  are  given  for  intervals  of  12  hours,  and 
we  assume  successively  £  =  0,  =^2,  —  •&»  e*c->  we  sna^  obtain 
the  hourly  motion  corresponding  to  each  hour  of  the  interval  for 
which  £,  c,  d,  etc.,  are  computed.  If  we  wish  the  hourly  motion 
for  the  instant  midway  between  two  values  of  £,  we  must  make 
t  =  j  ;  in  which  case  the  above  coefficient  of  c  becomes  0, 

a  fl         tt.          l 

— 2T> 

"  e       "         0, 

and  we  have  the  hourly  motion  equal  to 


where  b  and  d  represent  the  first  and  third  differences  corre- 
sponding to  the  instant  for  which  the  hourly  motion  is  required. 
Ex.  2.  Required  the  variation  of  the  moon's  right  ascension 
in  one  hour  of  longitude  for  the  instant  of  the  Greenwich  tran- 
sit, January  5,  1854,  from  the  following  data : 


MISCELLANEOUS   PROBLEMS.  213 


Date. 

Moon's  R.  A. 

1st  Diff. 

2d  Diff. 

3d  Diff. 

January  4,  U.  T. 

h,      m.          s. 

23  55  36.86 

m.           s. 

24  21.19 

i. 

s. 

L.  T. 

0  19  58.05 

-32.06 

» 

23  49.13 

+9.68 

«        5,  U.  T. 

0  43  47.18 

23  26.75 

-22.38 

+  9.23 

L.  T. 

1     7  13.93 

-13.15 

23  13.60 

«        6,U.T. 

1  30  27.53 

The  difference,  23m.  49.13s.,  corresponds  to  the  instant  mid- 
way between  the  lower  transit,  January  4th,  and  the  upper  tran- 
sit, January  5th.  By  interpolation  in  the  usual  manner,  we 
find  the  first  difference  corresponding  to  the  upper  transit,  Jan- 
uary 5th,  to  be 

b  =  23m.  36.76s., 
and  the  third  difference  for  the  same  instant  is 

d=+  9.45s. 
Subtracting  -^th  °f  ^  from  b,  we  have 

23m.  36.37s., 
which,  divided  by  12,  gives 

118.03s., 

which  is  the  motion  in  right  ascension  for  one  hour  of  longitude, 
corresponding  to  the  instant  of  Greenwich  transit,  January  5th. 

PROBLEM. 

(228.)  Two  hour  circles,  PA,  PB,  make  with  each  other  a 
small  angle  at  P,  and  from  any  point,  A,  in  one  of  them,  an 
arc,  AC,  of  a  great  circle  is  let  fall  perpendicularly  on  the 
other  ;  it  is  required  to  find  the  difference  of  decli- 
nation of  the  points  A  and  C. 

Let  P  be  the  pole  of  the  earth,  AB  a  parallel  of  dec- 
lination, and  AC  an  arc  of  a  great  circle  perpendicu- 
lar to  PB. 

Then,  in  the  triangle  APC, 

R.  cos.  APC  =  tang.  PC  cot.  PA, 

-P^      cot.  PA 

or  cot.  PC= — -. 

cos.  APC 

If  we  put  6= the  declination  of  the  point  A,  <J'=the  jj1 


214  PRACTICAL   ASTRONOMY. 

declination  of  the  point  C,  and  a  the  difference  of  right  ascension 
of  A  and  C,  we  shall  have 


(1) 


cos.  a 

from  which  the  declination  of  the  point  C  may  be  computed  ; 
but  the  computation  requires  a  table  of  tangents  extending  to 
single  seconds,  and  seven  places  of  decimals.     We  may  obtain 
a  more  convenient  formula  as  follows  : 
By  Trig.,  Art.  77, 

.       j.      tang,  a—  tang,  b 
tang,  (a  —  b)  =  -  —  —  -. 

1  +  tang,  a  tang,  b 

tang.  6 

--  tang.  6 

Hence          tang.  (d'-6)=  C°S>  a 


l  ,  tang.2  6 
cos.  a 
__tang.  6(1  —  cos.  a) 

cos.  a  +  tang.2  6 

Since  a  is  supposed  to  be  a  small  arc,  we  will  put  cos.  a  in  the 
denominator  equal  to  unity,  and  it  becomes 


tang.  («r-d)  =  teng-  6(l-cos'  a)=tang.  6  cos.2  ,5(1  -cos.  a) 

sec.2  6 

But      tang.  <J  cos.2  6=  sin.  (5  cos.  d=  —  ^  —  ,  by  Trig.,  Art.  73. 

0 

And  by  Trig.,  Art.  74, 

1  —  cos.  a  —  2  sin.2  ja. 

Hence  tang.  (6'  —  6)  =  sin.  26  sin.2  Ja. 

-  If  we  suppose  a  to  be  expressed  in  minutes,  and  6'—  6  in  sec- 
onds, we  may  put 

tang.  (6'-d)  =  (d'-d)  sin.  1", 

and  sin.  ^=|  sin.  l/  =  a(30  sin.  1"). 

Hence,  by  substitution,  we  obtain 

d'_<j=a2(900sin.  lx/sin.  2<5)  ......  (2) 

Example.  Suppose  d=29°,  and  a=90/  ;  it  is  required  to  find 
the  value  of  6'—  6. 

Solution.-—  By  formula  (1),       tang.  29°  =  9.7437520 

cos.  9(X  =  9.9998512 
tang.  29°  0'  30X/.0=  9.7439008 
Therefore  6'  -  6  =  30/x.O. 


CATALOGUES  OF  FIXED  STARS. 

By  formula  (2),  900  =  2.9542 

sin.  lx/=4.6856 

sin.  58°  =9.9284 

902  =  3.9085 

<5'_  <f=30".0  =4.4767 

In  this  manner  was  computed  Table  XVIII.  It  will  be  ob- 
served that  the  declination  of  the  point  C  is  always  greater  than 
that  of  A,  whether  it  be  north  or  south  of  the  equator.  This 
correction  is  applied  to  the  moon's  decimation  in  computing 
eclipses  and  occultations. 

CATALOGUES  OF  THE  FIXED  STARS. 

(229.)  The  first  individual  who  constructed  a  catalogue  of  the 
stars  was  Hipparchus,  who  flourished  between  160  and  135 
years  B.C.  This  catalogue  contains  the  longitude  and  latitude 
of  1028  stars,  and  is  preserved  in  Ptolemy's  Almagest. 

The  next  catalogue  of  stars  is  that  of  the  Tartar  prince  Ulugh 
Beigh.  This  catalogue  contains  the  places  of  1019  stars,  de- 
rived from  observations  made  at  Samarcand.  The  epoch  is  the 
year  1437  A.D. 

The  next  catalogue  is  that  of  Tycho  Brahe,  containing  the 
places  of  777  stars,  for  the  year  1600.  Kepler  subsequently  en- 
larged this  catalogue,  from  Tycho  Brahe's  observations,  to  1005 
stars,  and  published  it  in  the  year  1627. 

Halley's  catalogue  of  southern  stars  contains  the  places  of  341 
stars,  derived  from  observations  made  at  St.  Helena.  The  epoch 
is  1677.  This  was  the  first  catalogue  constructed  from  obser- 
vations made  by  the  use  of  telescopic  sights. 

The  catalogue  of  Hevelius  contained  1564  stars,  and  was  pub- 
lished in  1690.  The  epoch  is  1660. 

Flamsteed's  British  Catalogue  was  published  in  1725.  It 
contains  the  places  of  2935  stars,  reduced  to  the  year  1689. 

The  observations  made  by  Bradley  between  the  years  1750 
and  1762  were  published  by  Bessel  in  1818.  The  number  of 
stars  in  this  catalogue  is  3112.  The  epoch  is  1750. 

Lacaille's  catalogue  of  stars  in  the  southern  hemisphere,  ob- 
served at  the  Cape  of  (rood  Hope,  contains  9766  stars,  reduced 
to  the  year  1750.  This  catalogue  was  printed  in  1845  at  the 
expense  of  the  British  government. 


216  PRACTICAL    ASTRONOMY. 

Mayer  is  the  author  of  a  catalogue  of  998  zodiacal  stars,  pub- 
lished in  1775. 

(230.)  Toward  the  close  of  the  last  century  M.  L.  Lalande, 
at  Paris,  undertook  to  determine  the  positions  of  all  the  stars  in 
the  northern  hemisphere  down  to  the  ninth  magnitude.  These 
observations  have  recently  been  reduced  at  the  expense  of  the 
British  Association,  and  were  published  in  1847.  This  catalogue 
contains  the  places  of  47,390  stars,  reduced  to  the  year  1800. 

In  1814  Piazzi  published  a  catalogue  of  7646  stars,  from  ob- 
servations made  at  Palermo  from  1792  to  1813.  This  was  the 
most  important  catalogue  of  stars  hitherto  published.  Every 
star  was  observed  several  times,  and  a  mean  of  all  the  results 
taken  as  the  final  place  of  the  star. 

In  1806  De  Zach  published  a  catalogue  of  1830  zodiacal  stars, 
from  observations  made  by  him  at  Seeberg,  in  Saxe  Grotha. 

In  1838  was  published  Groombridge's  catalogue  of  4243  cir- 
cumpolar  stars,  reduced  to  the  year  1810.  It  contains  the  places 
of  all  the  stars  down  to  the  eighth  magnitude,  situated  within 
50°  of  the  north  pole,  derived  from  observations  made  between 
the  years  1806  and  1816. 

In  1821  Bessel  commenced  observations  of  all  the  stars  down 
to  the  ninth  magnitude,  comprehended  between  the  parallels  of 
15°  south  declination  and  45°  north  declination.  This  work 
was  completed  in  1833,  and  contained  about  75,000  observa- 
tions. Professor  Weisse  has  published  a  catalogue  of  31,895 
stars,  reduced  to  the  year  1825,  containing  all  the  stars  observed 
by  Bessel  within  the  region  extending  15°  on  each  side  of  the 
equator.  Professor  Weisse  is  at  present  engaged  in  reducing 
the  remaining  observations  of  Bessel. 

Argelander  has  observed  in  a  similar  manner  all  the  stars  in- 
cluded between  45°  and  80°  of  north  declination.  These  ob- 
servations were  commenced  in  1841,  and  were  finished  in  1844. 
The  number  of  stars  is  about  22,000.  Argelander  has  recently 
completed  a  similar  survey  of  the  heavens  between  15°  and  31° 
of  south  declination,  containing  17,600  stars. 

In  1821  Sir  Thomas  Brisbane  erected  an  observatory  at  Par- 
amatta, in  New  South  Wales,  and  in  1835  published  a  catalogue 
of  7385  stars,  chiefly  in  the  southern  hemisphere,  founded  upon 
observations  made  at  his  establishment. 


CATALOGUES  OF  FIXED  STARS.      217 

In  1844  was  published  Taylor's  catalogue  of  11,015  stars, 
founded  on  observations  made  at  Madras  during  the  years 
1822  to  1843. 

(231.)  In  1849  was  published  a  catalogue  of  2156  stars, 
formed  from  the  observations  made  during  12  years,  from  1836 
to  1847,  at  the  Royal  Observatory,  Greenwich. 

Professor  Rumker  has  recently  pompleted  a  catalogue  of  more 
than  12,000  stars,  observed  since  1836,  with  the  meridian  circle 
of  the  Hamburg  Observatory.  The  chief  object  in  preparing 
this  catalogue  was  to  supply  the  deficiencies  in  the  Histoire  Ce- 
leste and  Bessel's  Zones,  rather  than  to  supersede  the  catalogues 
previously  existing. 

Since  1838,  Mr.  Johnson,  director  of  the  RadclifTe  Observa- 
tory, Oxford,  has  been  engaged  in  reobserving  the  circumpolar 
stars  of  Grroombridge's  catalogue.  This  work  is  nearly  com- 
pleted, and  will  embrace  about  2000  stars  more  than  are  con- 
tained in  Grroombridge's  catalogue. 

Lieutenant  GHlliss,  director  of  the  United  States  astronomical 
expedition  to  Chili  in  1849,  undertook  to  construct  a  catalogue 
of  all  the  stars  down  to  the  eighth  magnitude,  situated  within 
60°  of  the  south  pole.  The  time  to  which  the  expedition  was 
limited  having  expired  in  1852,  this  plan  could  not  be  carried 
fully  into  execution ;  but  observations  were  obtained  of  nearly 
30,000  stars  within  the  25°  surrounding  the  pole. 

The  catalogue  published  by  the  British  Association  in  1845 
contains  the  places  of  8377  stars,  reduced  to  1850,  derived  from 
the  best  authorities,  and  furnishes  the  constants  for  deducing 
the  apparent  from  the  mean  places  with  the  greatest  facility. 
This  is  the  most  valuable  catalogue  for  general  use  which  has 
yet  been  published. 

In  1852,  Professor  Struve,  of  St.  Petersburg!!,  published  a  cat- 
alogue giving  the  places  of  2874  stars  for  the  year  1830,  being 
chiefly  double  stars  observed  at  Dorpat  from  1832  to  1843. 

The  catalogue  appended  to  this  volume  contains  the  mean 
places  of  all  stars  as  large  as  the  fifth  magnitude,  reduced  to 
the  year  1850.  The  mean  places  of  such  as  were  found  in 
Airy's  twelve-year  catalogue  were  taken  from  that  catalogue ; 
the  others  were  taken  from  the  catalogue  of  the  British  Asso- 
ciation, which  also  furnished  the  constants  for  reduction. 


218  PRACTICAL   ASTRONOMY. 

Determination  of  the  apparent  places  of  the  fixed  stars. 
(232.)  The  positions  of  the  stars  given  in  the  catalogues  are 
called  their  mean  places;  and  as  these  places  vary  from  year 
to  year  in  consequence  of  precession,  the  epoch  for  which  the 
places  are  given,  should  always  be  stated.  In  the  accompany- 
ing catalogue,  the  mean  places  of  the  stars  are  given  for  Janu- 
ary 1,  1850.  In  our  observations,  however,  we  do  not  find  the 
stars  in  the  positions  here  given  ;  hut  their  places  are  altered  by 
the  amount  of  their  precession  since  January  1,  1850,  and  are 
also  affected  by  aberration  and  nutation.  The  mean  places 
must  therefore  be  reduced  to  the  apparent  before  they  can  be 
compared  with  observation. 

The  algebraic  expressions  for  these  corrections  have  been  re- 
duced by  Bessel  to  the  following  form : 

Correction  in  R.  A. 

Correction  in  N.  P.  D. 

where  A=  — 18/7.732  cos.  O , 

B=-20".420sin.  O, 

C=^-0.025sin.  2O  -0.343  sin.  ft  +  0.004  sin.  2ft, 
T)=  -Ox/.545  cos.  2O  -9".250  cos.  ft  +0X/.090  cos.  2ft , 
a=+cos.  a  sec.  <5, 
6=  -fsin.  a  sec.  d, 

c=+  3.0706s.  +  1.3370s.  sin.  a  tang.  6, 
d=+co$.  a  tang.  d, 
a'—  —tang,  w  cos.  d+sin.  a  sin.  d, 
b'——  cos.  a  sin.  d, 
c'—  —  20".055  cos.  a, 
df—  +sin.  a. 
Also, 

£=the  time  from  the  beginning  of  the  year,  expressed 

in  fractional  parts  of  a  year. 
O  =the  sun's  true  longitude, 

ft  =the  mean  longitude  of  the  moon's  ascending  node, 
a  =  the  mean  right  ascension  of  the  star, 
6= the  mean  declination  of  the  star, 
« = the  obliquity  of  the  ecliptic. 

The  factors  A,  B,  C,  D  are  independent  of  the  star's  places, 
and  are  the  same  for  all  the  stars,  but  vary  with  the  time. 
They  are  given  for  every  day  of  the  year  in  the  English  Nau- 


CATALOGUES  OF  FIXED  STARS.      219 

tical  Almanac,  on  page  22  of  each  month,  and  on  pages  223-5 
of  the  American  Nautical  Almanac  for  1855.  The  factors  a,  6, 
c,  d,  a',  b',  c',  d',  depend  only  on  the  places  of  the  stars,  and  are 
sensibly  constant  for  a  long  period  of  years.  They  are  accord- 
ingly calculated,  and  their  logarithms  are  entered  opposite  each 
star  in  the  accompanying  catalogue. 

(233.)  In  order  to  obtain  the  correction  to  the  mean  place  of 
a  star,  we  have  only  to  take  from  the  catalogue,  and  opposite 
to  the  given  star,  the  logarithms  of  a,  £,  c,  d,  and  a',  b',  c',  d', 
with  their  proper  signs  ;  and  to  write  down  under  them  respect- 
ively, from  the  Nautical  Almanac,  opposite  the  given  day,  the 
logarithms  of  A,  B,  C,  D,  with  their  proper  signs,  remembering 
that  the  signs  prefixed  to  the  logarithms  affect  only  the  natural 
numbers.  We  then  add  each  pair  together,  and  find  the  nat- 
ural number  corresponding  to  the  sum.  The  sum  of  the  four 
natural  numbers  thus  obtained  (regard  being  had  to  their  signs) 
will  be  the  total  correction  required  in  right  ascension  and  polar 
distance  on  the  given  day.  This  correction,  applied  to  the  mean 
place  of  the  star  at  the  beginning  of  the  year,  will  give  the  ap- 
parent place  of  the  star  on  the  day  required. 

The  mean  right  ascension  and  polar  distance  for  the  epoch  of 
the  catalogue  is  reduced  to  that  of  the  current  year,  by  adding 
as  many  times  the  annual  precession  in  right  ascension  and 
polar  distance  as  the  number  of  whole  years  elapsed  since  the 
given  epoch. 

Example.  Required  the  apparent  right  ascension  and  north 
polar  distance  of  y  Orionis,  on  February  5,  1855,  for  midnight, 
at  Washington. 


220 


PRACTICAL   ASTRONOMY. 


Mean  R.  A.,  Janu- 
ary 1,  1850   ...  5 
5  years'  precession 
and  proper  motion 

771.                  S. 

17     5.31 
+  16.10 

Mean  P.  D.,  Janu- 
ary 1,  1850  .  .  . 
5  years'  precession 
and  proper  motion 
Mean  P.  D.,  Janu- 
ary 1,  1855   .  .  . 

Logarithms. 

a'     -9.5120 
A     -1.1366 

Of                    If 

83  47  27.7 
-18.6 

Mean  R.  A.,  Janu- 
ary 1,  1855   ...  5 

17  21.41 

83  47     9.1 

Logarithms. 

a     +8.0963 
A      -1.1366 

Nat.  Nos. 

-0.171 
+0.920 
-0.400 
-0.008 

Nat.  Nos. 

+4.452 
-0.281 
+0.465 

-6.134 
=  -1.498 

Aa     -9.2329 

Aa'     +0.6486 
V     -8.3039 
B     +1.1451 

b     +8.8188 
B     +1.1451 

B6     +9.9639 

B6'     -9.4490 

c     +0.5070 
C     -9.0952 

cr     -0.5721 
C     -9.0952 

Cc     -9.6022 

Cc'     +9.6673 

d     +7.1304 
D     -0.7954 

df     +9.9923 
D     -0.7954 

Dd     -7.9258 

Dd'     -0.7877 
Correction  of  P.  D. 

Correction  of  R.  A.  = 

+  0.341 

Hence  the  apparent  right  ascension  of  y  Orionis, 

=5h.  17m.  21.41s.  +  0.34s.  =  5h.  17m.  21.75s.; 
and  the  apparent  north  polar  distance, 

=  83°  47X  9//.l-l//.5  =  83°  47'  7x/.6. 

The  mean  right  ascension  on  the  1st  of  January  of  the  year 
of  observation  may  be  found  by  applying  to  the  apparent  ob- 
served right  ascension  the  above  correction  with  the  contrary 
sign. 

DIURNAL    ABERRATION    OF    LIGHT. 

(234.)  The  diurnal  aberration  of  light  is  a  phenomenon  result- 
ing from  the  movement  of  light,  combined  with  the  rotation  of 
the  earth  on  its  axis ;  and  it  differs  from  the  annual  aberration 
merely  in  consequence  of  the  difference  between  the  velocity  of 
the  earth  on  its  axis,  and  its  velocity  in  its  orbit. 

The  velocity  of  rotation  of  a  point  on  the  equator  is  to  the  ve- 
locity of  the  earth  in  its  orbit  as  1  to  65.82 ;  and,  since  the  an- 
nual aberration  is  20x/.445,  the  diurnal  aberration  will  be 


EQUATIONS   OF    CONDITION.  221 

Ox/.3107  in  arc, 
or  0.0207s.  in  sidereal  time. 

(235.)  This  is  the  diurnal  aberration  of  a  star  on  the  equator 
for  a  place  situated  on  the  equator ;  but  for  a  place  in  latitude 
0,  the  circle  of  diurnal  rotation  being  less  than  at  the  equator, 
in  the  ratio  of  radius  to  the  cosine  of  the  latitude,  the^aberration 
will  be  equal  to 

0.0207s.  cos.  0. 

If  the  star  be  not  in  the  equator,  this  expression  denotes  only 
the  aberration  on  the  parallel  of  the  star's  declination ;  and  in 
order  to  reduce  it  to  the  celestial  equator,  or  to  the  value  of  the 
aberration  in  right  ascension,  it  must  be  multiplied  by  the  se- 
cant of  the  star's  declination.  Hence,  if  </>  denote  the  latitude 
of  the  place,  and  6  the  declination  of  the  star,  then  the  correction 
in  time  for  the  upper  transit,  on  account  of  daily  aberration,  is 

+  0.0207s.  cos.  $  sec.  <J; 
and  for  the  lower  transit, 

—  0.0207s.  cos.  (f>  sec.  6. 

METHOD    OF    SOLVING    EQUATIONS    OF    CONDITION. 

(236.)  In  astronomical  researches  it  is  frequently  required  to 
determine  the  values  of  several  quantities  from  a  large  number 
of  simple  equations,  which  are  called  equations  of  condition  ; 
and  when  the  number  of  independent  equations  is  greater  than 
the  number  of  unknown  quantities,  these  equations  can  not  be 
perfectly  satisfied,  and  we  can  only  obtain  more  or  less  probable 
values  of  the  unknown  quantities.  The  given  equations  may 
be  combined  in  a  variety  of  ways,  and  each  mode  of  combina- 
tion will  furnish  different  values  of  the  unknown  quantities. 
Hence  it  is  a  question  of  the  highest  importance  to  determine 
in  what  manner  these  equations  should  be  combined,  so  as  to 
furnish  the  most  probable  values  of  the  unknown  quantities. 
Thus,  for  example,  if  our  observations  have  furnished  the  four 
equations, 

x-  y  +  2z  =  3 (1) 

5 (2) 

y+4z=21 (3) 

14: (4) 

and  it  is  required  to  find  the  values  of  x,  y,  and  z,  we  may  pur- 


222  PRACTICAL   ASTRONOMY. 

sue  various  methods,  and  we  shall  obtain  various  results  for 
these  quantities. 

(237.)  A  method  frequently  practiced  consists  in  rendering 
the  coefficients  of  one  of  the  unknown  quantities,  as  #,  positive  in 
all  the  equations,  and  then,  by  adding  all  the  equations  together, 
obtaining  •  new  equation,  in  which  the  coefficient  of  x  is  the 
greatest  possible.  In  a  similar  manner,  by  rendering  the  co- 
efficients of  y  positive  in  all  the  equations,  and  then  adding  all 
the  equations  together,  we  obtain  a  new  equation,  in  which  the 
coefficient  of  y  is  the  greatest  possible.  Proceeding  in  the  same 
manner  with  each  of  the  other  unknown  quantities,  we  shall 
have  as  many  new  equations  as  there  are  unknown  quantities, 
and  these  equations  may  be  readily  solved  by  the  ordinary  rules 
of  algebra.  Thus,  by  changing  the  signs  of  all  the  terms  in 
equation  (4),  and  adding  the  equations  together,  we  obtain 

9x-y-2z  =  15 (5) 

Changing  the  signs  in  equation  (1),  we  obtain,  in  the  same 
manner, 

5:r-f  77/  =  37 (6) 

Changing  the  signs  in  equation  (2),  we  obtain,  in  a  similar 
manner, 

Z  +  ?/-fl4*=:33 (7) 

From  equations  (5),  (6),  and  (7),  by  the  usual  method  of  elim- 
ination, we  obtain  the  values 

z  =  2.4853;  7/  =  3.5105;  *  =  1.9289. 

The  method  practiced  by  Tobias  Mayer  consisted  in  combining 
the  given  equations  by  addition,  subtraction,  etc.,  in  such  a  man- 
ner that  one  of  the  unknown  quantities,  as  x,  should  have  a  very 
large  coefficient  in  the  resulting  equation,  and  the  other  unknown 
quantities  should  have  .small  coefficients.  Another  combination 
would  furnish  a  final  equation,  in  which  only  y  should  have  a 
large  coefficient ;  and  so  for  each  of  the  unknown  quantities. 

(238.)  Methods  similar  to  the  preceding  are  frequently  used 
by  astronomers,  on  account  of  their  convenience ;  but  Legendre 
has  demonstrated  that  the  most  probable  values  of  the  unknown 
quantities  are  those  which  render  the  sum  of  the  squares  of  all 
the  errors  the  least  possible.  This  method  is  accordingly  called 
the  method  of  least  squares. 

If  we  substitute  in  equation  (1)  the  values  of  x,  y,  and  #, 


EQUATIONS    OF    CONDITION.  223 

above  given,  we  shall  find  that  the  second  member  of  the  equa- 

tion reduces  to  2.8326  instead  of  3,  showing  that  these  values 

do  not  perfectly  satisfy  the  equation.     A  similar  remark  applies 

to  equations  (2),  (3),  and  (4).      If,  then,  we  transpose  all  the 

terms  to  one  member  of  the  equation,  the  sum  of  the  terms  will 

not  reduce  to  zero,  but  will  be  equal  to  a  small  quantity,  e.     If 

these  equations  were  deduced  from  observation,  e  may  be  re- 

garded as  the  error  of  one  of  the  observations.     The  equations 

may  therefore  be  expressed  under  the  form 

e   =  a  +  bx    +  cy 

e'  =  of  +  b'x  +c'y 

e"  -  a"  -f  b"x  -f  c"y  +  d"z 

etc.  etc. 

There  will  be  as  many  of  these  equations  as  there  are  obser- 
vations. For  convenience,  let  us  denote  all  the  terms  of  the 
second  members  of  the  equations  which  are  independent  of  #, 
by  M,  W,  etc.,  and  we  shall  have 

e   -bx   +M 
ef  =  b'x  +  M' 


etc.       etc. 

Taking  the  sum  of  the  squares  of  these  equations,  we  shall 
have 

e2+e/2+e//2+,  etc.=(bx+M.)2+(b/x+W)2+(b//x+'M.//y+,  etc. 

(239.)  According  to  the  principle  above  stated,  this  quantity 
must  be  made  a  minimum,  which  is  done  by  putting  its  first  dif- 
ferential coefficient  equal  to  zero.  If  we  consider  only  the  un- 
known quantity  x9  we  shall  have,  after  differentiating  and  di- 
viding by  2dxj 

Q  =  b(bx  +  M.)  +  b/(b/x  +  W)-<rb//(b//x  +  'M.//),  etc.  ; 
that  is,  to  form  the  equation  that  gives  a  minimum  for  any  one 
of  the  unknown  quantities,  as  x,  we  must  multiply  each  equa- 
tion of  condition  by  the  coefficient  of  x  in  that  equation,  taken 
with  its  proper  sign,  and  put  the  sum  of  all  these  products 
equal  to  zero.  We  must  proceed  in  the  same  manner  for  y,  z9 
etc.,  and  we  shall  obtain  as  many  equations  of  the  first  degree 
as  there  are  unknown  quantities,  whose  values  may  then  be  ob- 
tained by  the  usual  mode  of  elimination. 

To  apply  this  method  to  the  example,  on  page  221,  we  must 


224  PRACTICAL   ASTRONOMY. 

multiply  equation  (1)  by  1  ;  equation  (2)  by  3  ;  equation  (3)  by 
4  ;  and  equation  (4)  by  —  1,  and  we  shall  obtain 
x-  y+  2z-   3  =  0 


x-3?j-  3^+14  =  0 

Putting  the  sum  of  these  equations  equal  to  zero,  we  obtain 
27x+6y-88  =  Q  .........  (8) 

We  must  now  multiply  equation  (1)  by  —  1  ;  equation  (2)  by 
2  ;  equation  (3)  by  1  ;  and  equation  (4)  by  3,  and  we  shall  obtain 
-x+  y—   2z  +   3  =  0 


7/4-  4^  —  21  =  0 
90-42  =  0 
Putting  the  sum  of  these  equations  equal  to  zero,  we  obtain 

6£+15y+;z-70  =  0  .  .-.«V>tf'V  .  .  (9) 

We  must  also  multiply  equation  (1)  by  2  ;  equation  (2)  by  —  5  ; 
equation  (3)  by  4  ;  and  equation  (4)  by  3,  and  we  shall  obtain 
2x~  2//  +   4z-   6  =  0 


Wx+  4//+16z-84  = 


Putting  the  sum  of  these  equations  equal  to  zero,  we  obtain 
7/  +  54*-107  =  0  ........  (10) 

By  comparing  equations  (8),  (9),  and  (10)  in  the  usual  mode 
of  elimination,  we  obtain  the  values 

.    z  =  2.4702;  #  =  3.5509;  *=1.9157. 

If  we  substitute  in  equations  (1),  (2),  (3),  and  (4),  of  page  221, 
the  values  found  in  Art.  237,  we  shall  find  the  errors  of  these 
several  equations  to  be 

-.1674,  -.1676,  +.1673,  -.1671; 
the  sum  of  whose  squares  is 

0.1120. 

If  we  substitute  in  the  same  equations  the  values  last  found, 
we  shall  find  the  errors  of  these  several  equations  to  be 

-.2493,  -.0661,  +.0945,  -.0704; 
the  sum  of  whose  squares  is 

0.0804. 
The  sum  of  the  squares  of  the  errors  resulting  from  employ- 


EQUATIONS   OF    CONDITION.  225 

ing  the  last  obtained  values  of  x9  y,  and  z9  is  less  than  that  re- 
sulting from  the  former  values ;  and  hence,  according  to  the 
principle  of  Legendre,  the  last  values  have  a  greater  probability 
in  their  favor  than  the  former. 

An  example  of  the  application  of  this  method  will  be  found 
on  page  334. 

P 


CHAPTER  X. 

ECLIPSES  01  THE  MOON. 

(240.)  THE  time  of  beginning  or  end  of  a  lunar  eclipse  at 
any  place,  may  be  found  by  adding  its  longitude  to  the  times 
given  in  the  Nautical  Almanac  for  the  meridian  of  Washington 
when  the  longitude  is  east,  or  subtracting  tHe  longitude  when 
it  is  west.  The  times  given  in  the  Nautical  Almanac  may  be 
deduced  from  the  right  ascensions,  declinations,  etc.,  of  the  sun 
and  moon,  by  the  following  method. 

An  eclipse  of  the  moon  can  only  happen  at  the  time  of  full 
moon.  If  the  moon  at  that  time  is  within  about  12  degrees  of 
one  of  its  nodes,  there  may  be  an  eclipse.  To  find  whether  there 
will  be  one,  and  to  calculate  the  times  and  phases,  proceed  as 
follows : 

(241.)  Find  the  "Washington  mean  time  of  opposition  in  right 
ascension  by  the  Nautical  Almanac,  in  the  manner  explained 
in  Art.  225.  For  this  time  compute  the  declination,  horizontal 
parallax,  and  semi-diameter,  both  of  the  sun  and  moon ;  also 
the  hourly  motion  of  the  moon  from  the  sun,  both  in  right  as- 
cension and  declination,  as  explained  in  Art.  226. 

Let  C  represent  the  centre  of  the  earth's  shadow  (see  figure 
on  page  opposite),  whose  right  ascension  is  the  same  as  that  of 
the  sun,  increased  by  12  hours,  and  its  declination  is  the  dec- 
imation of  the  sun,  with  a  contrary  sign.  Let  DCM  be  a  me- 
ridian passing  through  the  centre  of  the  shadow,  and  ACB  a 
great  circle  perpendicular  to  it.  Select  a  convenient  scale  of 
equal  parts,  and  from  it  take  CGr,  equal  to  the  moon's  declina- 
tion, minus  the  declination  of  the  centre  of  the  shadow,  and 
set  it  on  CD,  from  C  to  Gr,  above  the  line  AB,  if  the  centre  of 
the  moon  is  north  of  the  centre  of  the  shadow,  but  below  if 
south.  Take  CO,  equal  to  the  hourly  motion  of  the  moon  from 
the  sun  in  right  ascension,  reduced  to  the  arc  of  a  great  circle, 
and  set  it  on  the  line  CB,  to  the  right  of  C.  Take  CP,  equal 
to  the  moon's  hourly  motion  from  the  sun  in  declination,  and 


ECLIPSES   OF   THE    MOON. 


227 


set  it  on  the  line  CD,  from  C  to  P,  above  the  line  AB,  if  the 
moon  is  moving  northward  with  respect  to  the  shadow  ;  below, 


if  moving  southward.  Join  the  points  0  and  P.  The  line  OP 
will  represent  the  hourly  motion  of  the  moon  from  the  sun  ;  and 
parallel  to  it,  through  Gr,  draw  NGL,  which  will  represent  the 
relative  orbit  of  the  moon,  the  earth's  shadow  being  supposed 
stationary.  On  this  line  are  to  be  marked  the  places  of  the 
moon  before  and  after  opposition,  by  means  of  the  hourly  mo- 
tion OP,  in  such  a  manner  that  the  moment  of  opposition  may 
fall  exactly  on  the  point  Gr. 

(242.)  The  semi-diameter  of  the  earth's  shadow  is  equal  to 
the  horizontal  parallax  of  the  moon,  plus  that  of  the  sun,  minus 
the  sun's  semi-diameter;  which  result  must  be  increased  by 
-g^th  part,  on  account  of  the  earth's  atmosphere.  With  this 
radius  describe  the  circle  ADB  about  the  centre  C.  Add  the 
moon's  semi-diameter  to  the  radius  CB,  and  with  this  sum  for 
a  radius,  describe  about  the  centre  C  a  circle,  which,  if  there 
be  an  eclipse,  will  cut  NL  in  two  points,  E  and  H  representing 
respectively  the  places  of  the  moon's  centre  at  the  beginning 
and  end  of  the  eclipse.  Draw  the  line  CKR  perpendicular  to 
LN,  and  cutting  it  in  K.  The  hours  and  minutes  marked  on 


228  PRACTICAL   ASTRONOMY. 

the  line  LN,  at  the  points  E,  K,  and  H,  will  represent  respect- 
ively the  times  of  the  beginning  of  the  eclipse,  middle  of  the 
eclipse,  and  end  of  the  eclipse.  If  the  circle  does  not  intersect 
NL,  there  will  be  no  eclipse.  With  a  radius  equal  to  the  moon's 
semi-diameter,  describe  a  circle  about  each  of  the  centres,  E,  H, 
and  K.  If  the  eclipse  is  total,  the  whole  of  the  circle  about  K 
will  fall  within  ARE  ;  but  if  part  of  the  circle  falls  without 
ARE,  the  eclipse  will  be  partial.  In  either  case  the  magnitude 
of  the  eclipse  will  be  represented  by  the  ratio  of  the  obscured 
part,  RI,  to  the  moon's  diameter.  When  the  eclipse  is  total, 
the  beginning  and  end  of  total  darkness  may  be  found  by  taking 
a  radius  equal  to  CB,  diminished  by  the  moon's  semi-diameter, 
and  describing  with  it,  round  the  centre  C,  a  circle,  cutting  LN 
in  two  points,  representing  respectively  the  points  of  beginning 
and  end  of  total  darkness. 

Example  1. 

(243.)  Required  the  times  of  beginning,  end,  etc.,  of  the 
eclipse  of  the  moon,  October  24,  1855,  at  Washington  Observ- 
atory. 

By  the  Nautical  Almanac,  the  Washington  mean  time  of  op- 
position in  right  ascension  is,  October  24,  14h.  10m.  29.6s., 
which  result  differs  somewhat  from  that  found  on  page  211. 
Corresponding  to  this  time,  the  Nautical  Almanac  furnishes  the 
following  elements : 

o          >  // 

Declination  of  the  moon N.  11  42  26.9 

Declination  of  the  earth's  shadow N.  11  56  48.0 

Moon's  equatorial  horizontal  parallax  ....  59  45.8 

Sun's  horizontal  parallax 0    8.6 

Moon's  semi-diameter 16  19.2 

Sun's  semi-diameter 16    7.9 

Moon's  hourly  motion  in  right  ascension  .  .  33  22.1 

Sun's  hourly  motion  in  right  ascension  ...  2  23.4 

Hourly  motion  of  moon  in  declination  ...  N.  15  39.8 

Hourly  motion  of  shadow  in  declination   .  .  N.    0  52.1 


The  figure  of  the  earth  being  spheroidal,  that  of  the  shadow 
will  deviate  a  little  from  a  circle,  so  that  to  have  a  mean  ra- 


ECLIPSES    OF    THE    MOON. 


229 


dius  the  horizontal  parallax  of  the  moon  should  be  reduced  to  a 
mean  latitude  of  45°.  This  reduction,  by  Table  XIV.,  is  5/x.9  ; 
so  that  the  moon's  reduced  parallax  is  59'  39/x.9.  Then,  to  ob- 
tain CB,  the  semi-diameter  of  the  earth's  shadow,  we  have  59' 


39/x.9  +  8x/.6  -  16'  77/.9,  which  is  equal  to  43'  40".6.  Increasing 
this  by  -J^th  part  of  itself,  or  43/x.7,  we  have  447  24/x.3  =  CB ; 
to  which  adding  the  moon's  semi-diameter,  we  obtain  CE=60' 
43x/.5.  From  the  centre  C,  with  a  radius  CB,  taken  from  a  con- 
venient scale  of  equal  parts,  describe  the  circle  ARB,  represent- 
ing the  earth's  shadow.  Draw  the  line  ACB  to  represent  a  par- 
allel to  the  equator,  and  make  CGr  perpendicular  to  it,  equal  to 
147  21".l,  which  is  the  moon's  declination,  minus  the  declina- 
tion of  the  centre  of  the  shadow ;  the  point  Gr  being  taken  be- 
low C,  because  the  centre  of  the  moon  is  south  of  the  centre  of 
the  shadow. 

The  hourly  motion  of  the  moon  from  the  sun  in  right  ascen- 
sion is  30'  58x/.7,  which  must  be  reduced  to  the  arc  of  a  great 
circle  by  multiplying  it  by  the  cosine  of  the  moon's  declination, 
ll042/26//.9,Art.72,thus: 

30X  58".7  =  1858".7  =  3.269209 
cos.  Dec.=.  9.990870 

Reduced  hourly  motion  =  1820X/.0 = 3.260079 
Make  CO  equal  to  1820".0,  and  CP,  perpendicular  to  it,  equal 


230  PRACTICAL   ASTRONOMY. 

to  14'  47/x.7,  which  is  the  hourly  motion  of  the  moon  from  the 
shadow  in  declination,  the  point  P  being  placed  above  C,  because 
the  moon  was  moving  northward  with  respect  to  the  shadow. 
Join  OP ;  and  parallel  to  it,  through  Gr,  draw  the  line  NGrL, 
which  represents  the  path  of  the  moon  with  respect  to  the 
shadow.  On  NL  let  fall  the  perpendicular  CK.  Now  at  14h. 
10m.  29.6s.  the  moon's  centre  was  at  G-.  To  find  X,  the  place 
of  the  moon's  centre  at  14h.,  we  must  institute  the  proportion 

60m. :  10m.  29.6s. ::  OP  :  G-X  ; 

which  distance,  set  on  the  line  GrN,  to  the  right  of  Gr,  reaches 
to  the  point  X,  where  the  hour,  14h.  preceding  the  full  moon,  is 
to  be  marked.  Take  the  line  OP,  and  lay  it  from  14h.,  toward 
the  right  hand,  to  13h.,  and  successively  toward  the  left  to  15h., 
16h.,  etc.  Subdivide  these  lines  into  60  equal  parts,  represent- 
ing minutes,  if  the  scale  will  permit ;  and  the  times  correspond- 
ing to  the  points  E,  K,  and  H  will  represent  respectively  the  be- 
ginning of  the  eclipse,  12h.  36m. ;  the  middle  of  the  eclipse,  14h. 
22m. ;  and  the  end  of  the  eclipse,  16h.  7m. 

If  the  results  obtained  by  this  method  are  not  thought  to  be 
sufficiently  accurate,  we  may  institute  a  rigorous  computation. 

COMPUTATION    OF   ECLIPSE. 

(244.)  The  phases  of  the  eclipse  may  be  accurately  calculated 
in  the  following  manner  : 

In  the  right-angled  triangle  OOP,  we  have  given  CO  =  1820".0 
and  CP  =  S87/X.7,  to  find  OP  and  the  angle  CPO,  thus: 
CP:R::CO:tang.  CPO. 

CO  =  1820X/.0  =  3.260079 
CP=   887". 7  =  2.948266 
CPO  =  63°  59'  59"  tang.  =  0.311813 
Also,  sin.  CPO  :  R : :  CO  :  OP. 

CO  =  3.260079 

sin.  CPO  =  9.953659 

OP  =  2025/x.O  =  a306420 

The  angle  CPO  is  equal  to  CGrK,  because  GrE  and  OP  are 
parallel.  Then,  in  the  triangle  CGrE,  we  have  the  angle  CGE 
=  116°  Ox  l/x ;  CGr,  the  difference  of  declination  between  the 
moon  and  the  centre  of  the  shadow,  =  14'  21//.l  =  861//.l ;  and 
the  line  CE  =  60/  43//.5  =  3643//.5,  to  find  the  other  parts  of  the 
triangle,  thus : 


ECLIPSES    OF    THE    MOON.  231 

CE  :  sin.  CGE  ::  CG- :  sin.  CEG. 

CE  comp.  =  6.438481 

sin.  CGE  =  9.953659 

.       CG  =  2.935054 

CEG  =  12°  15X  51XX  sin  =  9.327194 

Therefore  the  angle  EGG- =  51°  44X  8/x.     Then 

sin.  CG-E:CE::sin.  EGG: EG-. 

sin.  CGE  comp.  =  0.046341 

CE  =  3.561519 

sin.  EGG  =  9.894959 

EG  =  3182x/.9  =  3^02819 

Then,  to  find  the  time  of  describing  EG,  we  say, 
As  OP  (2025/x.O)  is  to  1  hour,  so  is  EG  (3182XX.9)  to  the  time 
(5658.5s.)  Ih.  34m.  18.5s.,  between  the  beginning  of  the  eclipse 
and  the  time  of  opposition  in  right  ascension,  14h.  10m.  29.6s.. 
which  gives  the  beginning  of  the  eclipse  12h.  36m.  ll.ls. 

The  middle  of  the  eclipse  is  found  by  means  of  the  triangle 
CGK,  which  is  similar  to  CPO,  in  which  the  angles  and  hy- 
pothenuse  are  given  to  find  CK  and  KG.     We  have 
R  :  CG  ::  sin.  CGK  :  CK ::  cos.  CGK  :  GK. 
sin.  CGK  =  9.953659  cos.  CGK  =  9.641846 

CG  =  2.935054  CG  =  2.935054 

CK  =  774xx.O  ==  2.888713  GK  =  377'x.5  =  2.576900 

To  find  the  time  of  describing  GK,  we  form  the  proportion 

2025xx.O : 3600s. ::  377XX.5  : 671.1s.  =  llm.  ll.ls. ; 
Which  being  added  to  14h.  10m.  29.6s.,  because  the  point  K  falls 
to  the  left  of  G?  gives  the  time  of  the  middle  of  the  eclipse,  14h. 
21m.  40.7s.  Subtract  the  time  of  beginning,  12h.  36m.  ll.ls., 
from  the  time  of  middle,  we  obtain  for  half  the  duration  of  the 
eclipse  Ih.  45m.  29.6s. ;  which,  added  to  14h.  21rn.  40.7s.,  gives 
for  the  end  of  the  eclipse  16h.  7m.  10.3s. 

Subtracting  CK,  12X  54".0,  from  CR,  44X  24x/.3,  we  have  KR, 
31X  30XX.3 ;  to  which  adding  KI,  16'  19/x.2,  we  obtain  RI,  47' 
49/x.5.  Dividing  this  by  the  moon's  diameter,  32'  38XX.4,  we  ob- 
tain the  magnitude  of  the  eclipse,  1.465  (the  moon's  diameter 
being  unity) ;  and  the  eclipse  takes  place  on  the  moon's  north 
limb. 

(245.)  The  beginning  and  end  of  total  darkness  may  be  found 
in  the  same  manner.  With  a  radius  equal  to  CB,  diminished 


232  PRACTICAL   ASTRONOMY. 

by  the  moon's  semi-diameter  (that  is,  44X  24//.3-16/  19XX.2, 
which  equals  28X  5xx.l,  or  1685xx.l),  describe  about  the  centre 
C  a  circle,  cutting  LN  in  the  points  S  and  T,  which  will  repre- 
sent the  points  of  beginning  and  end  of  total  darkness. 

In  the  triangle  CGS,  CG-  =  861".l,  CS  =  1685".l,  and  the  an- 
gle CGrS  =  116°  Ox  lxx.  Hence  we  have 

CS:sin.  CG-S  ::  CO  :sin.  CSG-. 

CS  comp.  =  6.773374 

sin.  CGS  =  9.953659 

GG  =  2.935054 

CS(r=27°  20X  29/x  sin.  =  9.662087 
Therefore  the  angle  SCG  =  36°  39X  30X/.     Then 
sin.  CG-S  :  CS  ::  sin.  SCG- :  SG-. 

sin.  CG-S  comp.  =  0.046341 

CS=3.226626 

sin.  SCO  =  9.776005 

G-S  =  1119XX.4=3.048972 

Then,  to  find  the  time  of  describing  GrS,  we  say, 

2025xx.O  :  3600s. ::  1119XX.4  : 1990.0s.  =  33m.  10.0s. ; 
which,  being  subtracted  from  14h.  10m.  29.6s.,  gives  the  be- 
ginning of  total  darkness,  13h.  37m.  19.6s.  Subtracting  this 
from  the  time  of  middle,  we  obtain,  for  half  the  duration  of  total 
darkness,  44m.  21.1s.,  which,  added  to  14h.  21m.  40.7s.,  gives, 
for  the  end  of  total  darkness,  15h.  6m.  1.8s. 

(246.)  The  contacts  with  the  penumbra  may  be  found  in  a 
similar  manner.  The  semi-diameter  of  the  penumbra  is  equal 
to  the  semi-diameter  of  the  shadow,  plus  the  sun's  diameter,  or 
44"  24XX.3  +  32'  1 5XX.8  =  76X  40".l.  If  we  take  the  circle  ARE, 
in  the  figure  on  page  229,  to  represent  the  limits  of  the  penum- 
bra, CE  will  be  equal  to  76X  40XX.1  +  16X  19X/.2  =  92X  59X'.3. 
Then,  in  the  triangle  CGE,  we  have  given  the  angle  CGE 
=  116°  Ox  lx/,  CG-  =  861//.l,  and  CE  =  5579x/.3,  to  find  GE, 
thus :  CE  :  sin.  CGE  ::  CG  :  sin.  CEG. 

CE  comp.  =  6.253420 

sin.  CGE  =9.953659 

CG =2.935054 

CEG  =  7°  58X  25XX  sin.  =  9142133 
Therefore  the  angle  E CO  =  56°  lx  34X/.     Then 


ECLIPSES   OF  THE    MOON. 


233 


sin.  CGE  :  CE  ::  sin.  EGG  :  EG-. 

sin.  COE  comp.  =  0.046341 

CE=  3.746580 

sin.  EGG  =  9.918708 


To  find  the  time  of  describing  EG,  we  say, 

2025/x.0 : 3600s. ::  5147/x.9  : 9151.9s.  =  2h.  32m.  31.9s. , 
which,  subtracted  from  14h.  10m.  29.6s.,  gives  the  first  contact 
with  the  penumbra  at  llh.  37m.  57.7s.  Subtracting  the  time 
of  first  contact  from  the  middle  of  the  eclipse,  14h.  21m.  40.7s., 
we  have  for  half  the  duration,  2h.  43m.  43.0s. ;  which,  added 
to  14h.  21m.  40.7s.,  gives,  for  the  last  contact  with  the  penum- 
bra, I7h.  5m.  23.7s. 

The  results  thus  obtained  are  as  follows : 

h       m.     s. 

First  contact  with  the  penumbra  at .  .  11  37  58 

First  contact  with  the  umbra 12  36  11 

Beginning  of  total  eclipse 13  37  20      Mean  time 

Middle  of  the  eclipse 14  21  41  }          at 

End  of  total  eclipse 15    6    2     Washington. 

Last  contact  with  the  umbra 16    7  10 

Last  contact  with  the  penumbra  ...  17    5  24  > 

Magnitude  of  the  eclipse,  1.465  on  the  northern  limb. 

To  obtain  the  time  for  any  other  place,  we  have  only  to  add 
or  subtract  the  longitude.  For  Cambridge  Observatory,  whose 
longitude  is  23m.  41.5s.  east  of  Washington,  the  times  will  ac- 
cordingly be 

Jl.       TO.        S. 

First  contact  with  the  penumbra  at  .  .  12    1  39 

First  contact  with  the  umbra 12  59  53 

Beginning  of  total  eclipse 14    1    1      Mean  time 

Middle  of  the  eclipse 14  45  22  f         at 

End  of  total  eclipse 15  29  43     Cambridge. 

Last  contact  with  the  umbra 16  30  52 

Last  contact  with  the  penumbra  ....  17  29    5 

Ex.  2.  Compute  the  phases  of  the  eclipse  of  May  1,  1855,  for 
Cambridge  Observatory,  Longitude  23m.  41.5s.  east  of  Wash- 
ington, from  the  following  elements : 


234 


PRACTICAL   ASTRONOMY. 


Washington  mean  time  of  opposition  in 

right  ascension lOh.  49m.  10.1s. 

Declination  of  the  moon S.  15°     V 

Decimation  of  the  earth's  shadow  .         .  S.  15    11 


Moon's  equatorial  horizontal  parallax   . 

Sun's  horizontal  parallax 

Moon's  semi-diameter 

Sun's  semi-diameter 

Moon's  hourly  motion  in  right  ascension 
Sun's  hourly  motion  in  right  ascension 
Hourly  motion  of  moon  in  declination  . 
Hourly  motion  of  shadow  in  declination 

Ans. 


57 

0 

15 

15 

31 

2 

S.  13 
S.    0 


24X 
32 
9 
8 

35 
54 
34 
23 
10 
45 


.4. 
.0. 
.4. 
.5. 
.6. 
.1. 
.2. 
.2. 
.1. 
.1. 


First  contact  with  the  penumbra  at  ...     8  27.6 

First  contact  with  the  umbra 9  30.1 

Beginning  of  total  eclipse 10  32.7     Mean  time 

Middle  of  the  eclipse 11  20.8  i         at 

End  of  total  eclipse .  ....  12     9.0     Cambridge. 

Last  contact  with  the  umbra 13  11.5 

Last  contact  with  the  penumbra    ....  14  14.0 

Magnitude  of  the  eclipse,  1.549  on  the  southern  limb. 


CHAPTER  XL 

ECLIPSES  OF  THE  SUN  AND  OCCULTATIONS. 
SECTION   I. 

METHOD    OF    PROJECTING    SOLAR    ECLIPSES. 

(247.)  IN  order  to  ascertain  whether  a  solar  eclipse  will  be 
visible  at  a  particular  place,  and  if  so,  to  determine  its  general 
appearance,  we  will  suppose  the  spectator  to  be  placed  at  the 
centre  of  the  sun,  to  look  down  upon  the  earth,  and  see  the 
moon  passing  across  its  disk.  The  earth,  in  that  case,  must  ap- 
pear to  him  like  a  flat  circular  disk,  as  the  full  moon  does  to  us ; 
and,  on  account  of  the  obliquity  of  the  ecliptic,  the  position  of  the 
pole,  as  well  as  the  path  described  by  each  point  on  the  earth's 
surface  in  consequence  of  the  diurnal  motion,  must  vary  with 
the  season  of  the  year.  At  the  time  of  the  vernal  equinox,  the 
plane  of  the  equator  passes  through  the  sun;  the  poles  must 
therefore  appear  to  be  situated  upon  the  margin  of  the  disk,  and 
the  equator  inclined  23J  degrees  to  the  ecliptic,  as  in  Fig.  1, 
where  AB  represents  the  ecliptic,  H,  Hx  the  poles  of  the  ecliptic, 
EQ,  the  equator,  P,  Px  the  poles  of  the  equator,  and  DB,  AF  par- 
allels of  latitude,  which  appear  to  the  spectator  like  straight 

Fig.  I. 


lines.  At  the  autumnal  equinox  the  parallels  of  latitude  will 
also  appear  as  straight  lines,  but  the  poles  of  the  earth  will  lie 
on  the  opposite  side  of  the  poles  of  the  ecliptic,  as  represented  in 
Fig.  2.  At  the  summer  solstice  the  north  pole  of  the  earth  will 


236 


PRACTICAL   ASTRONOMY. 


occupy  the  position  indicated  by  P,  in  Fig.  3 ;  the  south  pole 
of  the  earth  will  be  invisible,  the  equator  will  occupy  the  posi- 
tion EMQ,  and  the  parallels  of  latitude  will  all  be  projected  into 


Fig.  3. 


ellipses.  At  the  winter  solstice  the  south  pole  of  the  earth  will 
be  seen  as  represented  at  Px,  in  Fig.  4 ;  the  north  pole  will  be 
invisible,  and  the  equator  will  occupy  the  position  EMQ,.  These 
different  cases  may  all  be  readily  illustrated  by  means  of  a  ter- 
restrial globe. 

(248.)  In  order  to  project  an  eclipse  of  the  sun,  we  must  first 
represent  the  earth  as  it  would  appear  to  a  spectator  on  the  sun 
at  the  time  proposed.  We  must  then  draw  the  parallel  of  lati- 
tude corresponding  to  the  place  for  which  the  phases  of  the 
eclipse  are  to  be  determined,  and  mark  upon  this  parallel  the 
position  of  the  given  place  for  the  different  hours  of  the  day. 
We  must  then  draw  the  moon's  apparent  path  across  the  earth's 
disk,  and  mark  the  points  which  it  occupies  at  each  hour  of  its 
transit.  We  must  then  find  that  point  of  the  moon's  path,  and 
the  point  in  the  path  of  the  spectator,  marked  with  the  same 
times,  which  are  at  the  least  distance  from  each  other.  This 
will  indicate  the  time  when  the  eclipse  is  greatest.  We  must 
find,  in  the  same  manner,  that  point  of  the  moon's  path,  and  that 
point  in  the  path  of  the  spectator,  which  are  marked  with  the 
same  hour,  and  whose  distance  from  each  other  is  equal  to  the 
sum  of  the  semi-diameters  of  the  sun  and  moon.  This  will  in- 
dicate the  time  of  beginning  or  end  of  the  eclipse.  This  method 
will  be  easily  understood  from  the  following  example ; 


ECLIPSES   OF   THE    SUN.  237 

Example. 

(249.)  Required  the  times  and  phases  of  the  eclipse  of  the 
sun,  May  26,  1854,  at  Boston,  latitude  42°  21X  28"  N.,  longi- 
tude 4h.  44m.  14s.  "W.  of  Greenwich. 

By  the  Nautical  Almanac,  the  time  of  new  moon  at  Green- 
wich is,  May  26,  8h.  47.1m.  mean  time,  corresponding  to  4h. 
2.9m.  mean  time  at  Boston ;  or  4h.  6.1m.  apparent  time,  the 
equation  of  time  being  +  3m.  15.4s. 

For  this  time,  the  elements  of  the  eclipse  are  as  follows : 

Sun's  longitude .  65°  12"  32". 

Sun's  declination 21°  II7  17XX  N. 

Moon's  latitude 21X  30/X  =  1290XX  N. 

Moon's  hourly  motion  in  longitude    ....  1S07XX. 

Sun's  hourly  motion  in  longitude 144XX. 

Moon's  hourly  motion  in  latitude   .  .  .  .  .  167XX. 

Moon's  equatorial  horizontal  parallax  .  .  .          54X  32XX.6. 

Sun's  equatorial  horizontal  parallax  ....  8XX.5. 

Moon's  true  semi-diameter 14X  53XX.5. 

Sun's  true  semi-diameter 15X  48x/.9. 

The  geocentric  latitude  of  Boston,  which  is  to  be  used  in  the 
following  projection,  is  42°  10X  Oxx. 

The  relative  positions  of  the  sun  and  moon  will  be  the  same 
if  we  attribute  to  the  moon  the  effect  of  the  difference  of  their 
parallaxes,  and  suppose  the  sun  to  remain  in  his  true  position. 
This  difference  is,  therefore,  the  relative  parallax,  or  that  which 
influences  the  relative  position  of  the  two  bodies.  The  moon's 
equatorial  horizontal  parallax  is  54X  32/x.6  ;  its  horizontal  paral- 
lax for  Boston  (Art.  210)  is  54X  27XX.6 ;  and  the  relative  paral- 
lax is  54X  19x/.l,  or  3259x/.l,  which  represents  the  apparent  semi- 
diameter  of  the  earth's  disk,  if  seen  at  the  distance  of  the  moon 
from  the  earth ;  while  14X  53/x.5  represents  the  moon's  apparent 
semi-diameter,  seen  from  the  same  distance.  These  numbers 
will,  therefore,  represent  their  relative  magnitude  when  seen  at 
any  distance.  Take,  therefore,  AC  (see  figure  on  next  page), 
equal  to  3259,  from  any  convenient  scale  of  equal  parts,  and  de- 
scribe the  semicircle  ADB  to  represent  the  northern  half  of  the 
earth's  disk  as  seen  from  the  sun,  and  draw  CD  perpendicular 
to  AB  for  the  axis  of  the  ecliptic.  Take  the  chord  of  23°  28X 


238 


PRACTICAL   ASTRONOMY. 

ri 


ECLIPSES   OF   THE    SUN.  239 

(equal  to  the  obliquity  of  the  ecliptic),  corresponding  to  the  ra- 
dius AC,  and  set  it  off  on  the  circle  ADB,  upon  each  side  of  D, 
to  E  and  H.  In  this  and  several  subsequent  cases,  when  a 
chord  or  sine  is  required  corresponding  to  a  particular  radius,  it 
is  most  conveniently  obtained  from  a  sector,  but  may  be  derived 
from  any  scale  of  chords  or  sines.  Draw  the  line  EH,  cutting 
CD  in  K.  By  comparing  figures  1,  2,  3,  and  4,  on  pages  235 
and  236,  it  will  be  perceived  that  the  pole  of  the  earth,  as  viewed 
from  the  sun,  will  appear  to  revolve  with  the  seasons  of  the  year 
through  the  line  HKE  ;  and  since  H  is  its  position  at  the  vernal 
equinox,  its  distance  at  any  time  from  H  will  be  equal  to  the 
versed  sine  of  the  sun's  longitude  ;  or  its  distance  from  the  sol- 
stice, K,  will  be  equal  to  the  sine  of  the  difference  between  the 
sun's  longitude  and  90°,  or  270°.  Take,  then,  the  sine  of  90° 
-65°  12X.5,  that  is,  the  sine  of  24°  47X.5  to  the  radius  EK,  and 
set  it  off  from  K  to  P,  which  will  be  the  place  of  the  pole  of  the 
earth.  Draw  CP,  and  produce  it  to  cut  the  circle  ADB  in  M. 
The  line  CP  represents  the  northern  half  of  the  earth's  axis. 

(250.)  We  wish  now  to  represent  the  parallel  of  latitude  of 
Boston,  or  the  path  of  Boston  on  the  earth's  disk,  as  seen  from 
the  sun.  If  the  latitude  of  the  place  were  just  equal  to  the  sun's 
declination,  the  sun  would  be  vertical  at  noon,  and  Boston  would 
be  seen  precisely  at  the  centre  of  the  disk  at  C  ;  but  since  the 
latitude  exceeds  the  sun's  declination  by  20°  59X,  Boston  must 
be  seen  that  distance  north  of  the  point  where  the  sun  is  verti- 
cal, which,  when  projected  on  the  disk,  becomes  the  sine  of  the 
arc,  measured  from  C  on  the  axis  CP.  Take,  then,  the  sine  of 
20°  59X  to  the  radius  AC,  and  set  it  off  from  C  to  12.  This 
point  will  be  the  apparent  position  of  Boston  at  noon. 

If  the  earth  were  transparent,  Boston  would  be  seen  at  mid- 
night somewhere  upon  the  line  CM,  and  north  of  the  point  12. 
The  point  antipodal  to  that  at  which  the  sun  is  vertical,  and 
which  also  would  be  seen  at  C,  is  as  many  degrees  south  of  the 
equator  as  the  sun's  declination  is  north.  Hence  the  distance 
of  Boston  from  this  point  at  midnight  must  be  equal  to  the  lati- 
tude of  the  place  added  to  the  sun's  declination,  which  amounts 
to  63°  21X.  With  the  radius  AC,  take  the  sine  of  63°  21X,  and 
set  it  off  from  C,  upon  the  line  CP,  to  S.  The  point  S  will  rep- 
resent the  apparent  place  of  Boston  at  midnight. 


240  PRACTICAL   ASTRONOMY. 

The  line  12,  S  is  the  shortest  diameter  of  the  ellipse  into  which 
the  parallel  of  latitude  appears  projected,  from  being  seen  ob- 
liquely. The  point  T,  midway  between  12  and  S,  is  its  centre ; 
and  the  line  6,  6  drawn  through  T,  perpendicular  to  CM,  is  its 
longest  diameter.  The  line  6,  6  not  being  shortened  by  being 
seen  obliquely,  will  appear  of  the  length  of  the  radius  of  the  par- 
allel, which  is  equal  to  the  cosine  of  the  latitude.  The  comple- 
ment of  the  latitude  of  Boston  is  47°  50' ;  and  setting  off  its 
sine  each  way  from  T  to  6  and  6,  we  find  the  extremities  of  the 
longest  diameter,  which  must  be  the  points  on  the  disk  where 
Boston  will  be  seen  at  six  o'clock  in  the  morning  and  at  six 
o'clock  in  the  evening. 

(251.)  The  position  of  Boston  at  any  other  hour  of  the  day 
may  be  found  as  follows :  "With  a  radius  equal  to  T,  6,  take  the 
sine  of  15°  (corresponding  to  one  hour),  and  set  it  off  on  each 
side  of  the  point  T  to  the  points  marked  15°.  In  the  same  man- 
ner, set  off  the  sines  of  30°,  45°,  60°,  and  75°.  Through  these 
points  draw  lines,  as  in  the  figure,  parallel  to  CM.  With  a  ra- 
dius equal  to  ST,  take  the  sine  of  75°,  and  set  it  off  on  the  line 
1, 11,  from  the  point  marked  15°,  above  and  below  the  line  6,  6. 
In  the  same  manner,  set  off  the  sines  of  60°,  45°,  30°,  and  15°, 
from  the  points  marked  30°,  45°,  60°,  and  75°.  The  points  1, 
2,  3,  etc.,  obtained  in  this  manner,  will  represent  the  situation 
of  Boston  at  those  hours,  and  an  ellipse  drawn  through  these 
points  will  represent  its  apparent  path.  The  hours  must  be 
marked  from  noon  toward  the  right,  in  succession,  round  the 
curve.  The  path  touches  the  circle  ADB  in  two  points,  repre- 
senting the  points  of  sunrising  and  sunsetting,  which,  in  the 
present  figure,  are  4J  A.M.  and  7|  P.M.  These  points  divide 
the  path  into  two  parts,  of  which  one  represents  the  path  by  day 
and  the  other  by  night. 

(252.)  We  wish  now  to  represent  the  moon's  apparent  path 
across  the  earth's  disk.  From  the  same  scale  upon  which  AC 
was  measured,  take  an  interval  equal  to  the  moon's  latitude, 
1290X/,  and  apply  it  on  CD,  from  C  to  G-,  above  the  line  ACB, 
because  the  moon's  latitude  is  north.  Take  CO,  equal  to  1663", 
the  hourly  motion  of  the  moon  from  the  sun  in  longitude,  and 
set  it  on  the  line  CB,  from  C  to  0.  Draw  OR  perpendicular  to 
CB,  and  make  it  equal  to  167",  the  moon's  hourly  motion  in  lat- 


ECLIPSES   OF   THE    SUN.  241 

itude,  and  set  it  above  the  line  ACB,  because  the  moon  is  going 
northward.  Draw  the  line  CR,  which  represents  the  hourly  mo- 
tion of  the  moon  from  the  sun  on  the  relative  orbit ;  and  paral- 
lel to  this  line,  draw  the  relative  orbit  of  the  moon,  LGN,  on 
which  are  to  be  marked  the  places  of  the  moon  before  and  after 
the  conjunction,  by  means  of  the  hourly  motion,  CR,  so  that  the 
moment  of  the  new  moon  at  Boston  may  fall  exactly  on  the 
point  G-,  where  the  new  moon  is  at  4h.  6m.  This  may  be  done 
by  instituting  the  proportion 

60m. :  the  line  CR ::  6m. :  the  line  G,  IV. 

This  distance  is  to  be  set  off  on  the  line  GL,  from  G,  toward 
the  left,  to  the  point  IY,  the  place  of  the  moon  at  four  o'clock. 
Then  the  distance  CR  being  taken  in  the  compasses,  and  set 
from  IV,  both  toward  the  right  and  left,  as  often  as  may  be  nec- 
essary, gives  the  places  of  the  moon's  centre  at  3,  4,  5,  6,  etc., 
o'clock,  by  apparent  time.  These  hours  may  be  divided  into 
60  equal  parts,  representing  minutes,  if  the  scale  be  taken  suffi- 
ciently large. 

(253.)  Find,  by  trials  with  a  pair  of  compasses,  two  points, 
one  on  the  moon's  path,  and  the  other  on  the  path  of  the  specta- 
tor, both  of  which  are  marked  with  the  same  times,  and  which 
are  at  the  least  distance  from  each  other.  That  time,  which  in 
the  present  case  is  5h.  44m.,  is  the  instant  when  the  eclipse  is 
greatest. 

The  appearance  of  the  moon,  as  projected  upon  the  earth's 
disk  at  any  hour,  may  be  shown  by  taking  its  semi-diameter, 
893x/.5,  and  with  this  radius  describing  a  circle,  whose  centre  is 
the  point  where  the  moon's  centre  will  be  at  the  time  proposed. 
The  figure  shows  the  appearance  of  the  moon  at  5h.  44m.  If, 
with  a  radius  equal  to  the  sun's  semi-diameter,  948x/.9,  we  de- 
scribe a  circle  whose  centre  is  the  position  of  Boston  at  the  same 
instant,  this  circle  will  represent  the  sun's  disk  at  the  middle  of 
the  eclipse.  The  moon's  semi-diameter  being  considerably  less 
than  the  sun's,  the  eclipse  is  seen  to  be  annular  at  Boston. 
Throughout  the  entire  tract  represented  as  covered  by  the  moon's 
disk,  the  sun's  centre  must  be  invisible ;  that  is,  along  the  par- 
allel of  latitude  of  Boston,  between  the  hours  3  and  7,  which 
amounts  to  more  than  60  degrees  of  longitude  ;  and  throughout 
a  much  larger  area,  some  portion  of  the  sun's  disk  will  be  con- 


242  PRACTICAL   ASTRONOMY. 

coaled.  The  extent  of  this  area  may  be  determined  by  describ- 
ing a  circle  with  the  same  centre,  and  a  radius  equal  to  the  sum 
of  the  radii  of  the  sun  and  moon. 

(254.)  The  eclipse  must  commence  at  Boston  as  soon  as  the 
disks  of  the  sun  and  moon  begin  to  interfere.  Take,  then,  from 
the  scale  of  equal  parts,  with  a  pair  of  compasses,  an  extent 
equal  to.  the  sum  of  the  semi-diameters  of  the  sun  and  moon, 
1842/x.4,  and,  beginning  near  L,  set  one  foot  on  the  moon's  path 
and  the  other  fqot  on  the  path  of  the  spectator,  and  move  them 
backward  and  forwaf$Bfi§l  both  the  points  fall  into  the  same 
hour  and  minute  in  both  paths.  This  will  indicate  the  begin- 
ning of  the  eclipse,  which,  in  the  present  case,  is  4h.  30m.  Do 
the  same  on  the  other  side  of  the  moon's  path,  and  the  end  of  the 
eclipse  will  be  found,  in  the  same  manner,  at  6h.  51m.  We 
have  thus  obtained  the  following  results  for  Boston : 

Apparent  Time.        Mean  Time. 
h.     m.  h.     m. 

Beginning  of  eclipse 4  30     =     4  27  P.M. 

Greatest  obscuration 5  44     =     541 

End  of  eclipse 6  51     =     6  48 

The  results  are  obtained  in  apparent  time,  because  the  points 
1,  2,  3,  etc.,  on  the  parallel  of  Boston,  correspond  to  apparent 
time,  and  the  places  of  the  moon  upon  its  relative  orbit  were 
also  determined  for  apparent  time. 

When  this  projection  is  carefully  made,  it  will  furnish  the 
times  of  beginning  and  end  within  one  or  two  minutes. 

By  drawing  different  parallels  of  latitude,  we  may  determine 
the  phases  of  the  eclipse  at  any  number  of  places  required. 

SECOND    METHOD    OF    PROJECTION. 

(255.)  In  the  preceding  projection  we  have  employed  the 
longitude  and  latitude  of  the  sun  and  moon,  as  well  as  their 
hourly  motions  in  longitude  and  latitude ;  but  the  projection  may 
be  made  with  about  equal  facility  by  employing  the  right  ascen- 
sion and  declination  of  these  bodies.  This  method  differs  from 
the  preceding  in  only  a  few  particulars. 

In  the  figure  on  the  opposite  page,  AC,  the  radius  of  the  cir- 
cle of  projection,  is  the  difference  of  the  horizontal  parallaxes  of 
the  sun  and  moon;  CD  is  a  meridian  or  circle  of  declination ; 


ECLIPSES   OF  THE    SUN. 


243 


244  PRACTICAL   ASTRONOMY. 

6,  12,  6  the  projection  of  the  parallel  of  latitude  of  the  place ; 
LGN  the  moon's  apparent  path;  CGr  the  difference  of  declina- 
tion of  the  sun  and  moon  at  the  instant  of  conjunction  in  right 
ascension ;  C12  the  sine  of  the  sun's  zenith  distance  at  noon ; 
and  6T  the  radius  of  the  parallel  of  latitude. 

The  hourly  motion's  of  the  moon  and  sun,  heing  given  in  right 
ascension,  must  be  multiplied  by  the  cosine  of  the  declination  to 
reduce  them  to  an  arc  of  a  great  circle,  by  Art.  72.  CO,  in  the 
figure,  represents  this  reduced  hourly  motion  of  the  moon  from 
the  sun,  and  OR  the  hourly  motion  of  the  moon  from  the  sun  in 
declination.  The  distance  CR  represents  the  moon's  relative 
hourly  motion  on  its  apparent  path. 

We  will  apply  this  method  to  the  eclipse  of  May  26, 1854,  for 
Boston. 

(256.)  By  the  Nautical  Almanac,  conjunction  in  right  ascen- 
sion takes  place  at  8h.  55m.  43.2s.,  Greenwich  mean  time,  cor- 
responding to  4h.  llm.  29s.  mean  time  at  Boston,  or  4h.  14m. 
44s.  apparent  time.  For  this  time  we  obtain  from  the  Almanac 

the  moon's  hourly  motion  in  right  ascension  .  .  31X  18x/.9. 
"   sun's  hourly  motion  in  right  ascension  ...     2'  31".8. 

Hence  the  hourly  motion  of  the  moon  from  the  sun  in  right 
ascension  is  28'  47".l,  which,  multiplied  by  the  cosine  of  the 
moon's  declination,  21°  33'  32",  is  1606".3.  The  other  elements 
are  taken  directly  from  the  Almanac,  and  are  as  follows : 

Elements  of  the  Eclipse. 

Conjunction  in  right  ascension,  Boston 

apparent  time,  May  26 4h.  14m.  44s.  P.M. 

R  =  radius  of  circle  of  projection  (see 

page  237) =  3259".!. 

Reduced  hourly  motion  of  moon  from 

sun  in  right  ascension 1606".  3. 

Moon's  hourly  motion  from  sun  in  dec- 
lination    461x/.4. 

Moon  north  of  sun 1335/x.O. 

Sum  of  semi  -  diameters  of  sun  and 

moon 1842x/.4. 

Difference  of  semi-diameters  of  sun  and 

moon  .  55".4. 


ECLIPSES   OF  THE    SUN.  245 

Take  AC,  equal  to  3259",  from  any  convenient  scale  of  equal 
parts,  and  describe  the  semicircle  ADB  to  represent  the  north- 
ern half  of  the  earth's  disk,  and  draw  CD  perpendicular  to  AB 
for  the  axis  of  the  earth.  Open  the  sector  to  the  radius  AC, 
and  take  the  sine  of  20°  59' =0'— tf,  and  set  it  off  from  C  to  12, 
on  the  line  CD.  This  point  will  he  the  apparent  position  of 
Boston  at  noon.  With  the  same  radius,  take  the  sine  of  63° 
21/  =  0/  +  (S,  and  set  it  off  from  C  to  S,  on  the  line  CD.  The 
point  S  will  represent  the  apparent  place  of  Boston  at  midnight. 
Bisect  the  line  12 S  in  T,  and  through  T  draw  6,  T,  6  perpen- 
dicular to  CD.  With  the  same  opening  of  the  sector  as  before, 
take  the  cosine  of  the  latitude  of  the  place,  42°  10',  and  set  it 
off  each  way  from  T  to  6  and  6.  These  will  be  the  points 
where  Boston  will  be  seen  at  six  o'clock  in  the  morning  and 
six  o'clock  in  the  evening.  The  apparent  path  of  Boston  across 
the  earth's  disk  must  now  be  represented  as  described  on  page 
240. 

(257.)  The  projection  of  the  parallel  of  Boston  may  be  effect- 
ed without  the  aid  of  a  sector,  by  first  computing  the  quantities 
C12,  CS,  and  T6. 

C12=R  sin.  (0'-d)  =  3259".l  sin.  20°  58"  43//  =  1167//. 
CS=R  sin.  (0/  +  <?)  =  3259//.l  sin.  63°  21'  17"  =  2913". 
T6=:R  cos.  0'  =  3259".!  cos.  42°  10'  0"=2416". 

These  quantities  may  now  be  set  off  from  the  same  scale  as 
AC,  without  the  aid  of  a  sector. 

Take  an  interval  equal  to  1335",  which  is  the  difference  of 
declinations  of  the  sun  and  moon,  and  set  it  off  on  CD  from  C 
to  Gr,  above  the  line  ACB,  because  the  moon  is  north  of  the  sun. 
Take  CO,  equal  to  1606",  the  reduced  hourly  motion  of  the 
moon  from  the  sun  in  right  ascension,  and  set  it  on  the  line  CB, 
from  C  to  0  ;  draw  OR  perpendicular  to  CB,  and  make  it  equal 
to  461,  the  moon's  hourly  motion  from  the  sun  in  declination. 
Draw  the  line  CR,  which  represents  the  hourly  motion  of  the 
moon  from  the  sun  on  the  relative  orbit ;  and  parallel  to  this 
line  draw  the  relative  orbit  of  the  moon,  LGrN.  At  the  in- 
stant of  conjunction  in  right  ascension,  the  moon's  centre  will 
be  at  Gr. 

(258.)  Measure  the  distance  CR  on  the  scale,  and  say,  as 
60m. :  CR : :  the  minutes  of  the  time  of  conjunction  :  the  dis- 


246  PRACTICAL   ASTRONOMY. 

tance  from  Gr  to  the  first  full  hour  point  to  the  left.     CR  is  found 
by  the  scale  to  be  1671/x. 

60m. :  1671 ::  14.7m. :  409". 

Take  409"  from  the  scale,  and  set  it  from  Gr  to  IV.  Take  the 
distance  CR,  and  set  it  from  IV,  along  the  moon's  path,  to  V, 
VI,  etc.,  and  divide  each  hour  into  ten-minute  spaces.  Take 
from  the  scale  the  sum  of  the  semi-diameters  of  the  sun  and 
moon,  and  running  the  left  foot  of  the  dividers  along  the  moon's 
path,  while  the  other  is  kept  on  the  ellipse,  notice  when  both 
stand  on  the  same  hour  space.  Subdivide  that  portion  of  the 
moon's  orbit  into  single  minute  spaces,  and  that  on  the  ellipse 
into  10  or  5  minute  spaces.  Do  the  same,  keeping  the  right 
foot  of  the  dividers  on  the  moon's  path,  and  subdivide  the  spaces 
in  like  manner.  Also,  notice  what  hour,  or  portion  of  an  hour, 
on  the  moon's  path  is  nearest  to  the  corresponding  hour  on  the 
ellipse,  and  subdivide  these  portions  in  the  same  way.  Apply- 
ing the  dividers  set  to  1842",  we  find  that  the  feet  stand  on  cor- 
responding divisions  when  the  left  foot  on  the  moon's  path  marks 
4h.  29m.  40s.,  and  also  when  the  right  foot  marks  6h.  50m.  35s. ; 
the  former  denoting  the  time  of  beginning,  and  the  latter  the 
time  of  ending  of  the  eclipse  at  Boston. 

Apply  one  side  of  a  small  square  to  the  moon's  path,  and 
move  it  along  until  the  other  side  cuts  the  same  hour  and  min- 
ute on  both  lines.  This  is  the  moment  of  nearest  approach  of 
centres,  which  is  at  5h.  44m.  10s.  The  distance  between  these 
corresponding  points,  measured  on  the  scale,  is  47",  which  is 
the  distance  of  the  centres  of  the  sun  and  moon  at  that  time. 
This  being  less  than  55".4,  the  difference  of  the  semi-diameters 
of  the  sun  and  moon,  shows  that  the  eclipse  at  Boston  will  be 
annular. 

(259.)  With  a  radius  equal  to  893",  from  the  point  5h.  44m. 
10s.  of  the  moon's  path  as  a  centre,  describe  a  circle  represent- 
ing the  moon's  disk.  With  the  corresponding  point  of  the  el- 
lipse as  a  centre,  and  a  radius  equal  to  949",  describe  a  second 
circle  to  represent  the  sun's  disk.  These  circles  will  exhibit  the 
phase  of  the  eclipse  at  the  moment  of  greatest  obscuration.  The 
figure  represents  the  visible  portion  of  the  sun  at  this  time  as  an 
unequal  ring,  extremely  narrow  on  its  northern  side.  With  the 
dividers  open  to  55",  the  times  of  formation  and  rupture  of  the 


ECLIPSES   OF   THE    SUN.  247 

ring  may  be  determined  in  the  same  manner  as  the  beginning 
and  end  of  the  eclipse. 

The  results  of  the  projection  are  as  follows : 

Apparent  Time.          Mean  Time. 
h.     m.      s.  h.     m.      s. 

Beginning  of  the  eclipse  at  Bos- 
ton, May  26,  1854 4  29  40  =  4  26  25  P.M. 

Greatest  obscuration 5  44  10  =  5  40  55 

End  of  eclipse 6  50  35  =  6  47  20 

In  a  working  projection,  for  determining  the  phases  of  an 
eclipse  for  a  particular  place,  it  is  not  necessary  to  describe  all 
the  lines  given  in  the  figure.  Thus,  in  the  present  example,  it 
was  only  necessary  to  draw  that  portion  of  the  path  of  Boston 
corresponding  to  the  three  hours  which  include  the  eclipse,  viz., 
from  4  to  7  P.M. ;  but  this  part  should  be  drawn  with  the  ut- 
most care.  So,  also,  it  is  only  necessary  to  draw  the  moon's 
path  for  the  same  hours ;  but  the  portions  corresponding  to  the 
times  of  beginning,  middle,  and  end  of  the  eclipse  should  be  sub- 
divided as  accurately  as  possible. 


SECTION   II. 

TO  CALCULATE  THE  BEGINNING  AND  END  OF  A  SOLAR  ECLIPSE  FOR 
A  PARTICULAR  PLACE. 

(260.)  The  method  of  projections  already  explained  will  suf- 
fice to  furnish  a  general  idea  of  the  phenomena  of  an  eclipse, 
and  also  the  approximate  times  of  the  phases  for  any  place  re- 
quired. A  more  accurate  result,  however,  is  frequently  needed. 
This  may  be  obtained  in  the  following  manner : 

Assume  any  two  convenient  times  near  the  supposed  begin- 
ning and  end  of  the  eclipse.  If  we  have  no  previous  knowledge 
of  these  phases,  we  may  assume  the  hour  before  and  the  hour 
after  the  time  of  apparent  conjunction.  The  computations  are 
most  conveniently  performed  when  the  assumed  times  are  even 
hours  of  the  meridian  for  which  the  ephemeris  is  computed. 
With  these  times  calculate  the  places  of  the  sun  and  moon,  and 
also  the  corresponding  parallaxes,  according  to  Arts.  211,  212. 
The  relative  positions  of  the  sun  and  moon  will  be  the  same,  if 
we  attribute  to  the  moon  the  effect  of  the  difference  of  the  par- 


248 


PRACTICAL   ASTRONOMY. 


allaxes  of  the  two  bodies,  and  suppose  the  sun  to  remain  in  his 
true  position.  This  difference  is,  therefore,  the  relative  paral- 
lax, or  that  which  influences  the  relative  position  of  the  two 
bodies.  The  difference  between  the  equatorial  parallaxes  of  the 
sun  and  moon  must  be  multiplied  by  the  radius  of  the  earth  for 
the  place  of  observation,  in  order  to  obtain  the  parallax  of  the 
place,  Art.  210.  These  parallaxes,  applied  to  the  right  ascen- 
sions and  declinations  of  the  moon  for  the  hours  proposed,  as 
given  in  the  Nautical  Almanac,  will  furnish  its  apparent  right 
ascensions  and  declinations.  Take  the  difference  between  the 
apparent  places  of  the  moon  and  sun,  and  reduce  the  differences 
of  right  ascension  to  seconds  of  arc  of  a  great  circle.  These  ap- 
parent positions  of  the  moon  with  respect  to  the  sun  will  furnish 
its  apparent  relative  orbit ;  and  the  contact  of  limbs  will  evi- 
dently take  place  when  the  apparent  distance  of  the  centres  be- 
comes equal  to  the  sum  of  the  sun's  semi-diameter  and  the  aug- 
mented semi-diameter  of  the  moon. 

(261.)  Let  S  represent  the  position  of  the  sun's  centre,  which 
we  will  suppose  to  remain  at  rest  throughout  the  eclipse ;  let 
SR  and  SP  represent  the  apparent  differences  of  right  ascension 
of  the  sun  and  moon  for  the  two  selected  hours,  one  preceding 
and  the  other  following  the  apparent  conjunction,  and  RM,  PM/ 


the  differences  of  declination.     Draw  the  line  MM',  and  it  will 
represent  the  relative  direction  of  the  moon's  motion.     Let  SN 


ECLIPSES   OF  THE    SUN.  249 

be  the  meridian  passing  through  S,  and  suppose  B  and  E  to  be 
the  positions  of  the  moon  at  the  times  of  beginning  and  ending 
of  the  partial  eclipse.  Draw  MDH  perpendicular  to  SN,  and  »SC 
perpendicular  to  BE. 

In  the  triangle  SDM,  we  have 

SD:  DM::  rad.:  tang.  DSM. 
Also,  sin.  DSM  :  DM : :  rad. :  SM. 

In  the  triangle  HMMX,  HM  represents  the  hourly  motion  in 
right  ascension,  reduced  to  the  arc  of  a  great  circle,  and  HMX 
the  hourly  motion  in  declination,  and  we  have 

HM  :  HM7 ::  rad. :  tang.  HMM'  or  NSC. 
Also,  cos.  HMM7 :  rad. : :  gM  :  MM7, 

which  is  the  hourly  motion  of  the  moon  in  its  relative  orbit. 
The  angle  MSC  =  MSD  +  DSC. 

Then,  in  the  triangle  MSC, 

rad.  :SM::cos.  MSC  :  SC. 

In  the  triangle  BSC,  BS  represents  the  sum  of  the  radii  of  the 
sun  and  moon,  and  we  have 

BS:SC::rad.:cos.  BSC. 
The  angle  BSM  =  BSC-MSC. 

Also,  ESM  =  BSC  +  MSC. 

Now,  in  the  triangles  BSM  and  ESM,  we  have 

sin.  SBM  :  SM  ::  sin.  BSM  :  BM, 
and  sin.  SEM  :  SM  ::  sin.  ESM :  EM. 

"RM 

Also,  the  time  of  describing  BM=         , 

MM7 

TT 1VT 

and  the  time  of  describing  EM  =  — — . 

The  time  of  describing  BM  being  subtracted  from  the  time 
when  the  moon's  centre  was  at  M,  will  furnish  the  instant  of 
beginning  of  the  eclipse ;  and  the  time  of  describing  EM  being 
added  to  the  time  when  the  moon  was  at  M,  will  furnish  the  in- 
stant of  ending. 

(262.)  Ex.  1.  Required  the  time  of  the  beginning  and  ending 
of  the  solar  eclipse  of  July  28,  1851,  at  Cambridge,  latitude  42° 
227  48"  N.,  longitude  4h.  44rn.  30s.  W.  of  Greenwich. 

The  time  of  new  moon,  July  28,  is  2h.  40m.  Greenwich  time ; 
but  as  the  sun  at  Cambridge  is  near  the  eastern  horizon,  the 


250 


PRACTICAL   ASTRONOMY. 


effect  of  parallax  will  be  to  accelerate  the  eclipse,  and  we  will 
therefore  select  for  our  two  hours  of  computation  Ih.  and  2h. 
Greenwich  time.  For  these  times  we  take  out  the  right  ascen- 
sions and  declinations  of  the  sun  and  moon  from  the  Nautical 
Almanac.  The  moon's  equatorial  horizontal  parallax  at  Ih.  is 
60'  28". 6 ;  the  sun's  horizontal  parallax  is  8/7.4  ;  difference,  60' 
20/7.2 ;  reduction  to  the  latitude  of  Cambridge,  5x/.5 ;  making 
the  relative  horizontal  parallax  for  Cambridge  60'  14x/.7.  At 
2h.  we  find  it  to  be  60/  15x/.6. 

The  moon's  hour  angle  from  the  meridian  is  equal  to  the  si- 
dereal time,  minus  the  moon's  true  right  ascension.  July  28, 
Ih.  at  Greenwich  corresponds  to  July  27,  20h.  15m.  30s.  mean 
time  at  Cambridge,  which,  converted  into  sidereal  time  by  Art. 
159,  is  4h.  37m.  53.13s.  Subtracting  this  from  the  moon's  right 
ascension,  we  obtain  the  hour  angle,  3h.  47m.  23.50s.,  or  56° 
50X  52x/.5.  In  the  same  manner,  the  hour  angle  at  2h.  is  found 
to  be  42°  27X  26/x.l.  With  these  data  we  compute  the  paral- 
laxes in  right  ascension  and  declination  by  Arts.  211  and  212. 
The  differences  of  right  ascension  are  reduced  to  seconds  of  arc 
of  a  great  circle  by  multiplying  them  by  15  x  cosine  of  the 
moon's  apparent  declination. 

According  to  Art.  228,  the  declination  of  the  point  M  is  not 
exactly  the  same  as  that  of  D  (MD  being  supposed  to  be  a  per- 
pendicular let  fall  on  the  meridian  NS).  From  Table  XVIII. 
we  find  the  correction  to  be  added  to  the  moon's  declination  at 
Ih.,  for  a  difference  of  right  ascension  of  37s.  is  Ox/.3  ;  and  at  2h., 
for  a  difference  of  right  ascension  of  1m.  18s.  is  l".l. 

Hence  we  obtain  the  following  results : 


For  Ih.  Greenwich  Time. 

For  2h.  Greenwich  Time. 

Moon's  true  place  

R.A. 
h.  m.       s. 
8  25  16.63 
2  40.24 

Dec. 

0            '              // 

19  58     9.4  N. 
30     4.9 

R.  A.  • 

h.  m        s. 
8  27  52.72 
2     9.51 

Dec. 

0            /               // 

19  52  40.0  N. 

27  15.5 

Moon's  parallax  
Moon's  apparent  place  .  .  . 
Sun's  place  

8  27  56.87 
8  28  34.17 

19  28     4.5 
19     4  50.2 

8  30     2.23 
8  28  44.00 

19  25  24.5 
19     4  15.8 

Difference  

37.30 
527.5 

23  14.3 
1394.6 

1   18.23 
1106.7 

21     8.7 
1269.8 

Reduced  to  seconds  of  arc 

Accordingly,  we  find  SR  represents  527/7.5 ;  RM,  1394/x.6 ; 
PS,  1106/x.7  ;  PMX,  1269x/.8.  The  hourly  motion  in  declination 
is  therefore  124/x.8 ;  and  that  in  right  ascension,  reduced  to  an 
arc  of  a  great  circle,  is  1634//.2. 


ECLIPSES   OF   THE    SUN. 


251 


Then,  in  the  triangle  SDM,  we  have 

1394".6  : 527".5 ::  1 :  tang.  20°  43'  8"=:DSM. 
Also,        0in.  DSM  :  527".5 : :  1 :  SM  =  1491//.0. 
In  the  triangle  HMM',  we  have 

1634".2  : 124//.8 ::  1 :  tang.  HMM'^40  22X  1". 
Also, 

cos.  HMM' :  1 ::  1634".2  :  MMX  =  1639X/.0= hourly  motion  of  the 
moon  in  its  relative  orbit. 

MSC:=MSD  +  DSC=:250  5'  9". 
In  the  triangle  MSC,  we  have 

1 : 1491".0 ::  cos.  MSC  :  SC  =  1350".4. 

The  moon's  semi-diameter  is  not  the  same  for  the  beginning 
and  end  of  the  eclipse  ;  but  for  a  first  approximation  we  will  sup- 
pose it  to  remain  unchanged,  and  will  compute  it  for  the  time 
Ih.  30m.,  which  we  find  to  be  16'  28/x.9.  The  augmentation 
for  altitude,  Art.  217,  is  llx/.8.  The  sun's  semi-diameter  is  15X 
46/x.5,  making  SB=:32/  27//.2  =  1947//.2.  Then 

1947/x.2  : 1350".4 ::  1 :  cos.  BSC  =  46°  5X  32". 
Therefore  BSM  =  21°    Ox  23", 

ESM  =  71°  10'  41". 
Then  sin.  SBM  :  1491".0 ::  sin.  BSM  :  BM  =  770".7, 

sin.  SEM  :  1491".0 ::  sin.  ESM :  EM=2035//.0. 
The  time  of  describing  BM=Oh.  28m.  13s. 
The  time  of  describing  EM  =  lh.  14m.  30s. 


252 


PRACTICAL   ASTRONOMY 


Subtracting  the  time  of  describing  BM  from  lh.,  and  adding 
the  time  of  describing  EM  to  lh.,  we  obtain  the  Greenwich  times 
of  beginning  and  ending ;  and,  subtracting  4h.  44m.  30s.,  we 
obtain  the  results  in  mean  time  of  Cambridge  ;  viz., 

Beginning  of  the  eclipse  at  .  .  7h.  47m.  17s.  )  Cambridge 

End  of  the  eclipse 9h.  30m.    Os.  )  mean  time. 

(263.)  Since  the  hour  angle  of  the  moon  is  subject  to  the  va- 
riation of  nearly  15°  per  hour,  the  effect  produced  by  parallax  is 
to  give  considerable  curvature  to  the  apparent  relative  orbit  of 
the  moon.  This  curvature  is  more  decided  when  the  eclipse 
takes  place  near  to  the  horizon.  Hence  the  preceding  results, 
deduced  by  supposing  the  portion  of  the  orbit  described  during 
the  eclipse  to  be  a  straight  line,  are  only  approximate.  It  is 
probable,  however,  that  they  are  correct  within  one  or  two  min- 
utes, and  this  may  be  considered  sufficient  for  the  purposes  of 
the  observer.  If  a  more  accurate  determination  is  required,  we 
must  repeat  the  computation  for  the  times  here  obtained  ;  and 
it  is  better  to  conduct  the  computations  for  the  ^ginning  and 
end  independently  of  each  other,  deriving  the  beginning  of  the 
eclipse  from  the  assumed  time  near  the  beginning,  and  the  end 
from  the  assumed  time  near  the  end.  For  convenience,  we  may 
omit  the  seconds,  and  repeat  the  computation  for  Oh.  32m.  and 
2h.  15m.  Greenwich  time.  For  these  times  we  look  out  the  places 
of  the  sun  and  moon  from  the  Almanac.  The  relative  horizon- 
tal parallax  at  Oh.  32m.  is  GO7  U".2  ;  at  2h.  15m.  is  60'  15x/.9. 
The  moon's  hour  angle  from  the  meridian  for  Oh.  32m.,  Green- 
wich time,  is  63°  33'  48/x.4 ;  for  2h.  15m.  it  is  38°  51"  34x/.2, 
from  which  we  obtain  the  parallaxes  as  below.  Proceeding  as 
in  the  former  case,  we  obtain  the  following  results : 


For  Oh.  32m.  Greenwich  Time. 

For  2h.  15m.  Greenwich  Time. 

Moon's  true  place 

R.A. 
h.  m.       s. 
8  24     3.76 
2  51.16 

Decs. 

0            '                '/ 

20     0  40.1  N. 
31  35.5 

R.A. 
h.  m.       s 
8  28  31.73 
2     0.43 

Dec. 

0            /              // 

19  51   16.  3  N. 
26  39.4 

Moon's  parallax                .  . 

Moon's  apparent  place  .  .  . 
Sun's  place 

2  26  54.92 
8  28  29.59 

19  29     4.6 
19     5     6.2 

8  30  32.16 
8  28  46.46 

19  24  36.9 
19     4     7.1 

Difference  

1  34.67 
1338.7 

23  58.4 
1439.8 

1  45.70 
1495.4 

20  29.8 
1231.5 

Reduced  to  seconds  of  arc 

The  motion  in  declination  for  lh.  43m.  is  208/x.3 ;  hence  the 
motion  for  lh.  is  121x/.3 ;  and  in  right  ascension  it  is  1650". 9  j 
which  values  differ  a  little  from  those  found  on  page  250. 


ECLIPSES   OF    THE    SUN.  253 

Let  SR  in  the  figure,  page  251,  represent  1338x/.7  ;  and  RM 
1439/x.8.  Then,  as  before,  we  shall  have 

1439/x.8  : 1338XX.7  ::  1 :  tang.  DSM^42°  54X  58", 

sin.  DSM  :  1338".7  ::  1 :  SM  =  1966xx.O. 
Also,  1231XX.5  : 1495XX.4 ::  1 :  tang.  DSMX  =  50°  31'  40X/, 

sin.  DSMX :  1495/x.4 ::  1 :  SMX  =  1937XX.2. 
1650". 9 : 121". 3  ::  1 :  tang.  HMMX=4°  12X  8/x, 
cos.  HMMX :  1 ::  1650XX.9  : 1655X. 4= hourly  motion  in  orbit. 
Therefore  MSC  =  47°    7X    6/x, 

MXSC  =  46°  19X32X/, 
1 : 1966xx.O ::  cos.  MSC  :  SC  =  1337x/.8. 

The  moon's  semi-diameter  at  Oh.  32rn.  is  16X  28x/.6  ;  the  aug- 
mentation for  altitude  is  9/x.3 ;  the  sun's  semi-diameter  is  15' 
46XX.5;  making  SB  ^1944XX.4. 

In  the  same  manner  we  obtain  SE  =  1949x/.l.     Then 
1944XX.4: 1337x/.8 ::  1 :  cos.  BSC=46°  31X  33XX. 
Therefore  BSM  =  0°  35'  33X/. 

Then  sin.  J^M  :  1966/x.O : :  sin.  BSM  :  BM = 29x/.6. 
The  time  of  describing  BM  =  64.3s. 
Also,  1949xx.l :  1337XX.8  ::  1  :cos.  ESC  =  46°  39X  24X/. 
Therefore  BSM7  =  0°  19X  52XX. 

sin.  SEMX :  1937XX.2  ::  sin.  ESMX :  EM'  =  16".3. 
The  time  of  describing  EMX  =  35.5s. 
Hence  the 

Beginning  of  the  eclipse  is  at  .  7h.  48m.  34s.  )  Cambridge 
End  of  the  eclipse  is  at  .  .  .  .  9h.  31m.  5s.  )  mean  time. 
(264.)  In  observing  the  beginning  of  a  solar  eclipse,  it  is  im- 
portant for  the  accuracy  of  the  observation  that  we  should  know 
on  what  part  of  the  sun's  limb  the  eclipse  will  begin.  This  is 
easily  found  by  means  of  the  diagram,  page  251.  The  angle 
NSB  is  the  angle  of  position  of  the  moon's  centre  from  the  north 
toward  the  west,  at  the  beginning  of  the  eclipse  ;  or,  if  we  es- 
timate the  angle  of  position  from  the  north  toward  the  east,  it 
will  be  360°  —  NSB.  Also,  the  angle  of  position  from  the  north 
toward  the  east,  at  the  end  of  the  eclipse,  is  NSE . 

But        NSB  =  CSB-CSN=46°.5-4°.2:=420.3, 
and  NSE  =  CSE  +  CSN=46°.6H-40.2^50°.8. 

Hence,  at  the  beginning  of  the  eclipse,  the  angle  of  the  moon's 
centre  from  the  north  toward  the  east  is  3 17°. 7. 


254 


PRACTICAL    ASTRONOMY. 


At  the  end,  the  angle  of  the  moon's  centre  from  the  north  to- 
ward the  east  is  50°.  8. 

(265.)  The  following  formulae  embody  the  preceding  princi- 
ples in  a  form  convenient  for  computation  : 

Put  x  —  SR  =  the  difference  of  apparent  right  ascension  "be- 
tween the  sun  and  moon  in  arc  of  a  great  circle, 
at  an  assumed  instant  ; 

=  the  difference  of  apparent  declination  at  the 
same  instant,  corrected  by  Art.  228  ; 


0=  the  angle  NSM  ; 
t=the  angle  HMM'^DSC  ; 
r=the  angle  BSC^ESC  ; 
c/=MH=the  hourly  variation  of  x; 
y/=HM/  =  the  hourly  variation  of  y  ; 

of  the  semi-diameters  of  sun  and  moon. 


Z  —  • 


X 


sin.  (3     cos.  (3 
tang.  4  =  |^, 

x, 


cos.  i 


=  hourly  motion 


in  relative  orbit, 


=  z  cos. 


cos.  7  =  —. 
A 

Angle  BSM=r-(/3+0, 

BM=-^— sin.{ 
cos.  y 

z 


cos.  7 


Time  of  describing  BM  = 


Time  of  describing  EM  = 


BM  cos.  i 


EM  cos.  i 
x, 


After  we  have  obtained  the  approximate  times  of  beginning 
and  ending,  if  the  greatest  accuracy  is  required,  we  must  repeat 
the  computation,  with  separate  values  of  A  for  beginning  and 
end,  as  was  done  in  the  last  example. 

(266.)  Ex.  2.  Required  the  time  of  beginning  and  end  of  the 
solar  eclipse  of  May  26,  1854,  at  Cambridge  Observatory. 

As  we  have  already  projected  this  eclipse,  we  shall  avail  our- 
selves of  the  approximate  knowledge  already  obtained,  and  shall 
assume  for  our  times  of  computation  9h.  10m.,  and  llh.  30m., 
Greenwich  mean  time.  For  these  times  we  take  the  places  of 


ECLIPSES   OF   THE    SUN. 


255 


the  sun  and  moon  from  the  Nautical  Almanac.  The  moon's 
equatorial  horizontal  parallax  at  9h.  10m.  is  54X  32X/. 5  ;  the  sun's 
horizontal  parallax  is  8/x.5 ;  difference,  54X  24/x.O ;  which,  re- 
duced to  the  latitude  of  Cambridge,  becomes  54X  19'x.l.  At 
llh.  30m.  we  find  it  to  be  54X  17/x.3. 

The  sidereal  time  at  Cambridge,  corresponding  to  9h.  10m. 
Greenwich  mean  time,  is  8h.  41m.  55.21s.  Hence  the  moon's 
hour  angle  is  4h.  28m.  17.98s.,  or  67°  4X  29XX.7.  The  hour  angle 
at  llh.  30m.  is  100°  57X  lxx.8.  "With  these  data  we  obtain  the 
parallaxes  as  below.  The  following  are  the  results : 


For  9h.  10m.  Greenwich  Time. 

For  llh.  30m.  Greenwich  Time. 

Moon's  true  place  

R.A. 
h.  m.       s. 
4  13  37.23 
2  40.23 

Dec. 

21  35  27.7  N. 
28  27.5 

R.A. 
h.   m.       s. 
4  18  30.09 
2  49.83 

Dec. 

0           '             " 

21  54    4.9  N. 
36  50.0 

Moon's  parallax   

Moon's  apparent  place   .  . 
Sun's  place  ,  

4  10  57.00 
4  13     9.83 

21     7     0.2 
21   11  22.9 

4  15  40.26 
4  13  33.46 

21   17  14.9 
21   12  22.9 

Difference  

2  12.83 
1858.7 

4  22.7 
259.4 

2     6.80 
1772.2 

4  52.0 
295.0 

Reduced  to  seconds  of  arc 

The  hourly  motion  in  declination  is  237/x.6,  and  that  in  right 
ascension  1556/x.l. 


Then,  in  the  triangle  SDM,  we  have 
259/x.4  :  1858XX.7  ::  1  :  tang. 
sin.  DSM  :  1858XX.7  :: 


=  82°  3X  18X/, 

=  1876XX.7. 


Also,  295xx.O  :  1772/x.2  ::  1  :  tang.  PMXS  =  80°  32X  57X/, 

sin.  PMXS  :  1772XX.2  ::  1  :  SMX=:1796XX.6. 
To  avoid  confusion,  the  lines  SM,  SM'  are  omitted  from  the 
figure,  but  are  to  be  supplied  as  on  page  248. 
In  the  triangle  HMMX,  we  have 


1556/xl  :  237XX.6  ::  1  :  tang. 


=  8°  40X53 


cos.  HMMX  :  1  ::  1556x/.l  :  1574".!  =  the  hourly  motion  in  orbit. 


256 


PRACTICAL   ASTRONOMY. 


Hence  MSC  =  89°  15' 49", 

MSC  =  89°  13' 50'. 
In  the  triangle  MSC, 

1 :  SM  =  1876x/.7 ::  cos.  MSC  :  SC  =  24//.l. 
The  moon's  semi-diameter  at  9h.  10m.  is  14'  53".5 ;  the  aug- 
mentation for  altitude  is  7/x.2 ;  the  sun's  semi-diameter  is  15' 
48x/.9;  making  SB  =  1849".6. 

In  the  same  manner  we  obtain  SE  =  1843".5. 
In  the  triangle  BSC, 

l849".6:24".l::l:ooB.  BSC=890'l5'  10". 
Hence  BSM  =  39". 

sin.  SBM :  SM  =  1876".7 ::  sin.  BSM :  BM  =  27".2. 
The  time  of  describing  BM  =  62.2s. 
In  the  triangle  ESC, 

1843".o :  24".l ::  1 :  cos.  ESC =89°  15'  1". 
Hence  ESM'^1' 11". 

sin.  SEW :  SM/=rl796//.6 ::  sin.  ESMX :  EM/=47//.3. 
The  time  of  describing  EMX  =  108.1s. 
Hence  the  eclipse  begins  at.  .  4h.  26m.  32s.  )  Cambridge 
"       ends     "  .  .  6h.  47m.  18s.  )  mean  time. 
At  the  beginning,  the  angle  of  the  moon's  centre  from  north 
toward  east  is  262°  47. 

At  the  end,  the  angle  of  the  moon's  centre  from  north  toward 
east  is  80°  34X. 

(267.)  As  the  computation  thus  far  indicates  that  this  eclipse 
will  be  annular,  it  is  important  to  determine  precisely  the  time 
of  formation,  and  also  of  the  rupture  of  the  ring.  In  doing  this, 
we  can  not  assume  that  the  moon's  path  from  9h.  10m.  to  llh. 


ECLIPSES   OF   THE    SUN. 


257 


30rn.  is  a  straight  line.  By  inspecting  Table  XVI.,  we  shall  see 
that  the  parallax  in  right  ascension  increases  with  the  hour 
angle  until  this  angle  becomes  six  hours ;  and  after  that  it  di- 
minishes. Now  at  the  middle  of  this  eclipse,  the  moon's  hour 
angle  is  very  nearly  6  hours ;  so  that  the  parallax  in  right  as- 
cension is  greater  for  the  middle  of  the  eclipse  than  for  either 
the  beginning  or  end.  We  must,  therefore,  make  an  independ- 
ent computation  for  a  time  near  to  the  middle  of  the  eclipse, 
which  we  will  assume  at  lOh.  20m.  Greenwich  time. 

Proceeding  as  heretofore,  we  find  the  moon's  relative  parallax, 
reduced  to  the  latitude  of  Cambridge,  to  be  547  18/x.2,  and  the 
moon's  hour  angle  84°  O7  47x/.5,  whence  we  obtain  the  following 
results : 


For  lOh.  20m.  Greenwich  Time. 

Moon's  true  place    

R.  A. 

h.     m.           s. 

4  16     3.54 
2  52.54 

Dec. 

21  44  50.5  N. 
32  34.3 

Moon's  parallax    

Moon's  apparent  place  .... 

4  13  11.00 
4  13  21.64 

21  12  16.2 
21  11  52.9 

10.64 

148.8 

23.3 
23.3 

Reduced  to  seconds  of  arc  .  . 

In  the  annexed  figure,  let  MA 
represent  a  portion  of  the  moon's 
relative  orbit  on  a  much  larger 
scale  than  the  former  figure ;  let 
SR  represent  148/7.8,  and  RM 
23//.3. 

Then,  in  the  triangle  SMR, 

23".3 : 148/x.8 ::  1 :  tang.  SMR =81°  6'  V, 

sin.  SMR :  148/x.8 ::  1 :  SM  =  150x/.6. 
The  angle       MSC  =  SMR  -  ASC  =  72°  25'  8". 
1 :  SM ::  cos.  MSC  :  SC=45".5, 
1 :  SM ::  sin.  MSC  :  CM  =  143//.6. 
The  time  of  describing  CM  =  328.3s. 

Hence  the  nearest  approach  of  the  centres  of  the  sun  and  moon 
is  at  5h.  40m.  58.3s.  Cambridge  mean  time. 

The  semi-diameter  of  the  sun  is  15'  48//.9 ;  the  augmented 
semi-diameter  of  the  moon  is  14'  57x/.6 ;  difference,  51/x.3.  The 
least  distance  between  the  centres  of  the  sun  and  moon  is  45x/.5, 

R 


258  PRACTICAL   ASTRONOMY. 

Hence  the  eclipse  will  be  annular.  To  find  the  times  of  forma- 
tion and  rupture  of  the  ring,  with  S  as  a  centre,  and  a  radius 
equal  to  51x/.3,  describe  an  arc,  cutting  the  moon's  path  in  the 
points  B  and  E ,  which  will  represent  the  points  required. 

Then,  in  the  triangle  SCB, 

SB  :  1 : :  SC  :  cos.  BSC  =  27°  32", 
1:  SB::  sin.  BSC  :BC  =  23".7. 

The  time  of  describing  BC  =  54.2s. 

Hence  the 

Formation  of  the  ring  is  at .  .  5h.  40m.    4s.  )  Cambridge 

Rupture  of  the  ring  is  at  ...  5h.  41m.  52s.  )  mean  time. 

The  preceding  computations  were  all  in  type  in  1853,  but 
owing  to  the  destruction  of  the  stereotype  plates  by  fire  in  De- 
cember of  that  year,  it  became  necessary  to  re-cast  the  entire 
volume,  and  thus  its  publication  has  been  delayed  until  after 
the  occurrence  of  the  eclipse.  The  eclipse  could  not  be  observed 
at  Cambridge  on  account  of  the  interference  of  clouds.  The  in- 
stants of  first  and  last  contact  observed  at  New  York  and  Wash- 
ington differed  but  a  few  seconds  from  the  time  computed  from 
the  Tables. 


SECTION    III. 

OCCULTATIONS    OF    STARS    BY    THE    MOON. 

(268.)  Occultations  of  stars  by  the  moon  may  be  computed 
in  the  same  manner  as  eclipses  of  the  sun,  the  only  difference 
in  the  operation  consisting  in  this,  that  the  star  has  neither  mo- 
tion, parallax,  nor  semi-diameter.  These  circumstances  render 
the  computation  of  an  occupation  more  simple  than  that  of  an 
eclipse. 

Ex.  1.  It  is  required  to  find  the  times  of  immersion  and 
emersion  of  a  Tauri,  Jan.  23,  1850,  at  Cambridge  Observatory, 
latitude  42°  22X  48/x,  longitude  4h.  44m.  30s.  W.  of  Greenwich. 

The  Greenwich  mean  time  of  apparent  conjunction,  according 
to  the  Nautical  Almanac,  is  12h.  41m.  49s.  We  will,  therefore, 
select  12h.  and  13h.  as  the  two  hours  of  computation  for  the  first 
approximation. 

For  these  times  we  find  the  following  data : 


OcCULTATIONS     OF     STARS. 


259 


For  12h.  Greenwich  Time. 

For  13h.  Greenwich  Time. 

Moon's  true  place  

R.  A. 
h.  m.       s. 
4  25  36.22 
47.06 

Dec. 

0            /                >/ 

16  30    4.1  N. 
26  42.7 

R.A. 
h.  m.       s. 
4  28     4.53 
0.47 
4~28     5^00 
4  27  19.54 

Dec. 

0            /               // 

16  36  33.2  N. 
26  13.2 

Moon's  parallax  
Moon's  apparent  place  .  .  . 
Sun's  place  

4  26  23.28 
4  27  19.54 

16     3  21.4 
16  12     3.4 

16  10  20.0 
16  12     3.4 

Difference  
Reduced  to  seconds  of  arc 

56.26 
811.0 

8  42.0 
521.6 

45.46 
654.9 

1  43.4 
103.1 

The  moon's  horizontal  parallax,  reduced  to  the  latitude  of 
Cambridge  at  12h.,  is  59'  44/x.6 ;  at  13h.  it  is  59X  46/x.6.  The 
moon's  hour  angle  at  12h  is  14°  34'  4/x.8  east ;  at  13h.  it  is  0° 
8X  41x/.5  east  of  the  meridian,  from  which  we  compute  the  par- 
allaxes as  above. 

The  hourly  motion  in  right  ascension  is  1465/x.9,  and  that  in 
declination  418x/.5. 

Let  S  represent  the  position  of  the  star.  Take  SR  =  811".0, 
RM  =  521x/.6  ;  then  M  will  be  the  position  of  the  moon's  centre 
at  12h.  Take  SP  =  654//.9,  PM'  =  103".l ;  then  M'  will  be  the 
position  of  the  moon  at  13h.,  and  MM'  is  the  moon's  relative  orbit. 


Then,  in  the  triangle  SDM,  we  have 

521".6 :  811".0 ::  1 :  tang.  DSM  =  57°  15'  9", 

sin.  D  SM :  811".0 : :  1 :  SM  =  964/x.3. 
In  the  triangle  HMM', 

1465/7.9  : 418".5 ::  1 :  tang.  HMMX  =  15°  56'  1", 

cos.  HMMX :  1 ::  1465x/.9  :  MMX  = 
the  hourly  motion  of  the  moon  in  its  orbit. 


PRACTICAL   ASTRONOMY. 


Hence  MSC  =  73°  11"  10". 

In  the  triangle  MSC, 

1 : 964/x.3 ::  cos.  MSC  :  SC  =  278x/.9. 

The  radius  of  the  moon  at  12h.  30m.  is  978/x.6 ;  the  aug- 
mentation for  altitude  is  15x/.3 ;  making  BS  =  993".9. 
In  the  triangle  BSC, 

993".9 : 278x/.9 : :  1 :  cos.  BSC  =  73°  42'  7". 
Hence  BSM=     0°  30' 57", 

ESM  =  146°53/17//, 

sin.  SBM  :  964".3 ::  sin.  BSM :  BM  =  30".9, 
sin.  SEM :  964".3 ::  sin.  ESM :  EM  =  1876".8. 
The  time  of  describing  BM  =  1m.  13s. 
The  time  of  describing  EM^lh.  13m.  52s. 
Hence  the 

Immersion  takes  place  at ...  7h.  14m.  17s.  )  Cambridge 
Emersion  takes  place  at   ...  8h.  29m.  22s.  )  mean  time. 
(269.)  As  one  of  the  times  selected  for  computation  was  very 
near  the  instant  of  immersion,  it  is  probable  that  the  preceding 
result  for  immersion  is  pretty  accurate.     For  the  sake  of  verifi- 
cation, we  will,  however,  repeat  the  entire  computation  for  12h., 
and  13h.  15m.,  Greenwich  mean  time. 


OCCULTATIONS     OF     STARS. 


261 


For  12h.  Greenwich  Time. 

For  13h.  15m.  Greenwich  Time. 

Moon's  true  place 

R.A. 

h.  m.       s. 
4  25  36.22 
47.06 

Dec. 

0           /              li 

16  30    4.  IN. 

26  42.7 

R.A. 

h.  m.       s. 
4  28  41.66 
11.31 

Dec. 

16  38     9.5  N 
26  13.2 

Moon's  apparent  place  .  .  . 
Sun's  place  

4  26  23.28 
4  27  19.54 

16     3  21.4 
16  12     3.4 

4  28  30.35 
4  27  19.54 

16  11  56.3 

16  12     3.4 

Difference 

56.26 
811.0 

8  42.0 
521.6 

1   10.81 
1019.9 

7.1 
6.3 

Reduced  to  seconds  of  arc 

The  moon's  horizontal  parallax,  reduced  to  the  latitude  of 
Cambridge  at  13h.  15m.,  is  59/  47/x.O ;  and  the  moon's  hour  an- 
gle is  3°  27X  38XX.4  west,  from  which  we  obtain  the  parallaxes 
as  above.  The  hourly  motion  in  right  ascension  is  1464x/.7,  and 
in  declination  412XX.2. 

Hence,  in  the  triangle  HMMX, 

1464XX.7  : 412/x.2  ::  1 :  tang.  HMMX^  15°  43X  9X/, 

cos.  HMMX :  1 : :  1464XX.7 :  MMX  =  1521x/.6, 
the  hourly  motion  of  the  moon  in  its  orbit. 
Hence  MSC  =  72°  58X  18X/, 

1 : 964XX.3 ::  cos.  MSC  :  SC  =  282XX.4. 

The  radius  of  the  moon  at  12h.  is  978x/.3 ;  at  13h.  15m.  is 
979/x.O.  The  augmentation  at  12h.  is  15xx.l ;  at  13h.  15m.  is 
15XX.5.  Hence  SB  =  993XX.4,  and  SE  =  994/x.5. 

993x/.4 : 282XX.4 : :  1 :  cos.  BSC  =  73°  29X  6X/. 
Hence  BSM=0°  30X  48/x, 

sin.  SBM  :  964/x.3 ::  sin.  BSM  :  BM=30x/.4. 
The  time  of  describing  BM  =  71.9s. 

6XX.3 : 1019x/.9 ::  1 :  tang.  DSMX  =  89°  38X  46", 

sin.  DSMX :  1019XX.9 ::  1 :  SMX  =  1019XX.9. 
Hence  MXSC  =  73°  55X  37XX, 

994/x.5 : 282XX.4 ::  1 :  cos.  ESC  =  73°  30X 14". 
Hence  ESM'^00  25X  23XX, 

sin.  SEMX :  1019XX.9 ::  sin.  ESM" :  EM/=26//.5. 
The  time  of  describing  EMX  =  62.7s. 
Hence  the 

Immersion  takes  place  at  ...  7h.  14m.  18s.  )  Cambridge 
Emersion  takes  place  at    ...  8h.  29m.  27s.  )  mean  time, 
which  results  are  almost  identical  with  those  first  obtained. 

The  angle  of  position  of  the  point  S,  referred  to  the  moon's 
centre  at  immersion,  and  measured  from  the  north  toward  east, 
is  KBS,  which  equals  DSB  or  CSB-CSD. 


262  PRACTICAL   ASTRONOMY. 

The  angle  of  position  of  the  point  S  at  emersion,  measured 
from  the  north  toward  west,  is  LES,  which  equals  DSE,  or  CSE 
4-  CSD.  But  if  the  angle  he  measured  from  north  toward  east, 
which  is  the  usual  method,  it  is  360°  — DSE. 
«•  Hence  at  immersion  the  angle  of  position  of  the  star  is  58° 
from  the  north  point  of  the  moon's  limb. 

At  emersion  the  angle  of  position  of  the  star  is  271°  from  the 
north  point. 

(270.)  These  results  are  doubtless  correct  within  one  or  two 
seconds,  according  to  the  moon's  places  given  in  the  Tables ;  but 
it  is  not  to  be  supposed  that  these  times  are  absolutely  reliable 
to  this  degree  of  accuracy.  Burckhardt's  tables  of  the  moon 
frequently  exhibit  errors  of  15/x,  and  occasionally  of  30XX.  Now 
an  error  of  30/x  in  the  moon's  place  would  cause  an  error  of  more 
than  one  minute  in  the  computed  time  of  occultation.  However 
accurately,  therefore,  the  computations  are  performed,  the  result 
may  be  found  erroneous  by  half  a  minute  of  time,  and  occasion- 
ally even  more  than  a  minute.  For  simple  purposes  of  observa- 
tion, therefore,  there  is  little  advantage  in  making  the  computa- 
tions with  the  precision  which  is  here  attempted,  and  we  may 
generally  be  content  with  the  results  of  the  first  approximation. 
Indeed,  if  we  take  the  parallaxes  directly  from  a  table,  like 
Table  XVI.,  and  make  a  careful  geometrical  construction  with 
scale  and  dividers,  we  may  generally  obtain  the  time  of  begin- 
ning and  end  of  the  occultation  within  a  minute  of  the  truth, 
which  is  quite  sufficient  to  guide  the  astronomer  in  observing  an 
immersion.  For  an  emersion,  it  is  desirable  to  know  the  time 
as  accurately  as  possible,  in  order  that  the  eye  of  the  observer 
may  not  be  fatigued  by  too  long  watching  for  the  phenome- 
non. 

Ex.  2.  It  is  required  to  find  the  time  of  immersion  and  emer- 
sion of  j  Virginis,  January  9,  1855,  at  Washington  Observatory, 
latitude  38°  53X  39"  N.,  longitude  5h.  8m.  11s.  W.  of  Greenwich, 
from  the  following  data : 


Jan.  10,  Oh.  Gr.  Mean  Time.!  Jan  10,  In.  Gr.  Mean  Time. 

Moon's  right  ascension  .  . 
Moon's  declination    .  .  .  . 
Moon's  equatorial  hor.  par. 
Moon's  true  semi-diameter 

12h.  35m.  12.33s. 
0°    5X  32XX.3  S. 
55X  32XX.8 
15X  10/x.l 

12h.  37m.  2.80s. 
0°  19X  36/x.2  S. 
55X  34XX.4 
15X  10XX.6 

ECLIPSES    OF   THE    SUN. 


Right  ascension  of  y  Virginis,  12h.  34m.  18.43s. ;  decimation, 
0°  39'  12".l  south. 

Sidereal  time  of  mean  noon  at  Washington,  January  9,  19h. 
14m.  40.33s. 

Ans.  Immersion,  18h.  17m.  34s.  Washington  mean  time. 
Emersion,    19h.  36m.  45s.  "  "        " 

Angle  of  position  of  star  116°,  from  north  point  toward  east, 
at  immersion. 

Angle  of  position  of  star  322°,  from  north  point  toward  east, 
at  emersion. 


SECTION    IY. 
BESSEL'S  METHOD  OF  COMPUTING  SOLAR  ECLIPSES. 

(271.)  Bessel  has  developed  the  complete  theory  of  eclipses  in 
the  second  volume  of  his  Astronomical  Researches.  We  propose 
to  exhibit  the  main  points  of  this  theory,  together  with  its  appli- 
cation to  the  determination  of  geographical  longitudes. 

Let  S  represent  the  centre  of  the  sun,  M  that  of  the  moon,  E 


Fig.  1 


that  of  the  earth,  and  0  the  place  of  the  observer  on  the  earth's 
surface.  The  limbs  of  the  sun  and  moon  will  appear  to  be  in 
contact  when  the  point  0  is  situated  on  the  surface  of  the  cone 
which  circumscribes  these  two  bodies.  There  are  two  such  cir- 
cumscribing cones.  One  of  them,  YTVX,  has  its  vertex  at  T, 
between  the  centres  of  the  sun  and  moon  ;  the  other,  VT/V/,  has 
its  vertex,  T',  in  the  prolongation  of  the  line  MS,  on  the  side  of 


264  PRACTICAL   ASTRONOMY. 

Fig.  2. 


the  moon  which  is  opposite  to  the  sun.  If  the  point  0  is  situa- 
ted on  the  surface  of  the  first  cone,  an  observer  at  0  will  witness 
the  external  contact  of  the  disks  of  the  sun  and  moon ;  but  if  0 
is  on  the  surface  of  the  second  cone,  the  observer  will  see  the  in- 
ternal contact  of  the  disks. 

(272.)  In  order  to  obtain  the  equation  of  this  conical  surface, 
let  us  conceive  a  system  of  three  rectangular  axes,  whose  origin 
is  at  E,  the  centre  of  the  earth.  Let  the  axis  of  z,  or  EZ,  be 
drawn  parallel  to  the  line  MS,  which  joins  the  centres  of  the  sun 
and  moon.  We  will  assume  that  the  positive  direction  of  this 
line  is  that  which  proceeds  from  the  moon  to  the  sun,  and  also 
that  the  positive  end  of  the  axis  EZ  corresponds  to  a  point  of  the 
celestial  sphere  whose  right  ascension  is  A  and  declination  D. 
Also,  we  will  suppose  that  the  axis  of  #,  or  E  Y,  lies  in  the  plane 
which  passes  through  EZ  and  the  north  pole  of  the  equator,  and 
that  the  positive  end  of  this  axis  is  directed  toward  a  point  of 
the  celestial  sphere  whose  right  ascension  is  A,  and  whose  dec- 
lination is  90°  +  D.  The  third  axis,  or  the  axis  of  x,  is  the  line 
EX,  which  is  perpendicular  to  the  plane  of  the  hour  circle  ZE  Y, 
and  lies  in  the  equator,  at  the  distance  of  90°  from  the  intersec- 
tion of  the  equator  with  the  hour  circle  ZE  Y.  The  declination 
of  each  end  of  this  axis  will  be  zero ;  and  for  its  positive  direc- 
tion, we  will  assume  that  which  corresponds  to  a  right  ascension 
of  90°  + A.  We  accordingly  assume  that,  when  referred  to  the 
centre  of  the  earth, 

The  co-ordinates  Determine  the  position 

x,  y,  and  z,       of  the  centre,  M,  of  the  moon  ; 

x',  y' ',  and  z',     of  the  centre,  S,  of  the  sun ; 

|,    ?7,   and  £,       of  the  point  0,  or  the  place  of  the  observer. 


ECLIPSES   OF  THE    SUN.  265 

Also,  let  Gr=the  line  MS,  or  the  distance  from  the  moon  to 

the  sun  ; 
"        /=the  angle  OTM,  or  OT'M,  which  the  axis  of  the 

cone  forms  with  its  side  ; 
"         s=  the  perpendicular  distance  of  the  vertex  of  the 

cone  from  the  plane  YEX. 
Since  the  axis  EZ  is  parallel  to  the  line  MS,  we  have 

x'=x,  y'=y,  and  z'  =  z+G  ......  (1) 

(273.)  If  now,  from  the  vertex  of  the  cone  T,  and  from  the 
point  0,  we  draw  the  lines  TQ,  and  OB  perpendicular  to  the 
plane  XE  Y  ;  also  the  lines  Q,L  and  BI  in  this  plane,  perpendic- 
ular to  the  axis  EY  ;  and  the  line  IR,  parallel  to  the  line  BQ,, 
we  shall  have 


IL=y-i7,  and  RL 
Draw  the  plane  NOK  parallel  to  the  plane  YEX,  and  passing 
through  0,  the  place  of  the  observer.  In  this  plane  draw  the 
lines  ON,  OH  parallel  with  the  axes  EY  and  EX  ;  let  the  line 
MS  produced  meet  the  plane  OHN  in  K,  and  draw  KN  perpen- 
dicular to  ON.  Then  we  shall  have 


In  the  triangle  TOK,  right-angled  at  K,  we  have 

OK" 

tang.  /-tang.  OTK=^±;  and  TK 
JLlv 


Therefore      (X-SY+(y-rif  =  (S-tf  tang.2/    ....  (2) 

This  equation  corresponds  to  the  conical  surface  in  the  case 
of  an  external  contact.  A  similar  one  may  be  deduced  for  the 
conical  shadow  in  the  case  of  an  internal  contact. 

(274.)  Since  both  the  sun  and  moon  are  sensibly  spherical,  we 
may  represent  the  radius  of  the  moon  by  &,  and  that  of  the  sun 
by  k'.  Then,  from  the  similar  triangles,  MTW  and  STY,  right- 
angled  at  "W  and  V,  we  shall  have 

ST:SV::MT:MW. 

Also,  ST  +  TM  =  G. 

And  ST.  sin.  STY  =  SY, 

MT.sin.  MTW=MW. 

But       STY^MTM^/;  ST  =  ^-s;  WV  =  s-z. 

Consequently, 

z'—S'.kfr.s-s:k;  and  G-  sin.  /=  kf  +  k. 


266  PRACTICAL   ASTRONOMY. 

Consider  now  the  conical  surface  which  corresponds  to  the  in- 
ternal contact,  and  whose  vertex  is  at  Tx,  Fig.  2.     In  this  case 
we  shall  have 
T'M  =  QM-Q,T'  =  *--s;  T'S=*'-s,  and  MS  =  T'S-T'M  =  G. 

But 

T'S .  sin.  VT'S  =  SV  =  K;  TXM .  sin.  WT'M  =  MW = k. 

Consequently,  in  the  case  of  an  internal  contact,  we  shall 

have 

z'-sik'llz-S'.k;  and  G  sm.f=k/-k. 

Hence,  by  reduction,  we  obtain  for  an  external  contact, 

zkf+z'k        ,    .      ,    kf  +  k 
s  —  ~         — ,  and  sin.  f=—r. — . 
kf  +  k  G 

Also,  for  an  internal  contact, 

zkf—z'k  .  k'-k 

-- 


(275.)  It  remains  to  consider  in  what  cases  the  angle  /  is 
acute,  and  when  obtuse.  For  an  observer  at  the  point  0,  on 
the  earth's  surface,  the  vertex  of  the  conical  shadow  may  be 
situated  either  on  the  same  side  of  the  heavens  as  the  eclipsing 
body,  or  on  the  opposite  side.  The  first  case  always  happens  at 
an  external  contact,  and  also  at  an  internal  contact  in  an  annu- 
lar eclipse.  The  vertex  of  the  conical  shadow  is  then  found  ei- 
ther at  T,  Fig.  1,  or  at  Tx/,  Fig.  3.  The  second  case  happens 

Fig.  3. 


V 

when  the  eclipse  is  total,  and  the  contact  an  internal  one ;  in 
which  case  Tx,  Fig.  2,  is  situated  on  the  side  of  the  observer, 
which  is  opposite  to  the  sun.  If,  then,  we  reckon  the  angle  /, 
which  the  axis  of  the  cone  forms  with  its  side,  always  in  the 
same  direction,  we  shall  have  /=OTQ,,  Fig.  1,  or  /=OT"Q,, 
both  of  which  angles  are  acute;  and/^OT'Q,,  Fig.  2,  which 
is  an  obtuse  angle.  Hence  we  see  that  for  an  external  contact 
the  angle /is  always  acute,  and  also  fdf  an  internal  contact  in 
annular  eclipses;  but  for  an  internal  contact  in  total  eclipses 
this  angle  is  obtuse. 


ECLIPSES    OF  THE    SUN.  267 

(276.)  "We  will  now  eliminate  5  and  tang./  from  equation  (2), 
by  employing  the  values  just  found. 

.  (p     /£/     M2 

cos.2f=l—sin2f=-      v          ;  , 


cos.2/     ^- 
Hence 


or  ^  -Bffl  .....  (3) 

where  the  sign  +  belongs  to  the  external,  and  —  to  the  internal 
contacts. 

For  convenience,  let  us  put 


=tans.  f 


By  substituting  z  +  G  for  z'  in  equation  (3),  we  obtain 

(x-tf+(*J-^--=(l-itf  .......  (5) 

Comparing  this  equation  with  equation  (2),  we  see  that 

l  =  s  tang./; 

and  this  represents  the  radius  of  the  circle  formed  by  the  in- 
tersection of  the  conical  shadow  with  the  plane  which  passes 
through  the  centre  of  the  earth,  and  perpendicular  to  the  axis 
of*. 

(277.)  We  will  now  show  how  the  values  of  x,  £,  «/,  ??,  /,  z,  and 
£  may  be  computed  with  the  assistance  of  an  ephemeris.  For 
this  purpose,  conceive  a  new  system  of  rectangular  axes  inter- 
secting each  other  at  the  centre  of  the  earth.  Let  EZ,  the  new 
axis  of  z,  be  directed  toward  the  north  pole  of  the  equator  ;  let 
EX,  the  new  axis  of  re,  be  situated  in  the  equator,  and  directed 
toward  a  point  of  the  heavens  whose  right  ascension,  a!,  is  equal 
to  that  of  the  sun  from  the  earth.  Let  E  Y,  the  axis  of  ?/,  be 
directed  toward  a  point  of  the  equator  whose  right  ascension  is 
90°  +  a7;  and  also,  let  these  directions  correspond  to  the  posi- 
tive side  of  the  co-ordinate  axes.  Let  a,  d,  and  r  represent  the 
true  right  ascension,  declination,  and  the  distance  of  the  moon's 


268 


PRACTICAL   ASTRONOMY. 


centre  from  that  of  the  earth  ;  and  let  a',  dx,  and  R  represent 
the  same  quantities  for  the  sun's  centre,  where  r  and  R  are  sup- 
posed to  refer  to  the  same  unit  of  length. 

Let  M  represent  the  centre  of 
the  moon,  and  from  it  let  fall  upon 
the  plane  XEY  the  perpendicular 
MN=rjsr.  From  its  extremity  N, 
upon  the  line  EX,  let  fall  the  per- 
pendicular  NR=y,  and  represent 
ER  by  x. 

Then,  in  the  triangle  EMN, 
right-angled  at  N,  the  side  EM 
=  r  ;  MN  =  z  ;  and  the  angle 
MEN,  which  represents  the  in- 
clination of  the  line  EM  to  the 
equator,  is  —d. 

Her^ce  EN=r  cos.  6]  and  z—  r  sin.  6. 

Also,  in  the  triangle  ENR,  right  angled  at  R,  the  angle 


Hence 

NR=y=EN  sin.  (a  —  a')  =  r  cos.  6  sin.  (a  —  a')  ; 
and  ER=a;  =  r  cos.  6  cos.  (a—  ax). 

That  is,  we  find  the  co-ordinates  of  the  moon's  centre, 
parallel  to  the  new  axis  of  z,  to  be  r  sin.  c5  ; 

"  "  ?/,      "     r  cos.  6  sin.  (a  —  ax)  ; 

"     •         "  x,      "     r  cos.  6  cos.  (a—  ax). 

In  the  same  manner,  since  the  axis  of  x  has  the  same  right 
ascension  as  the  sun's  centre,  the  co-ordinates  of  the  sun 
parallel  to  the  axis  of  s  will  be  R  sin.  6'  ; 


x 


R  cos.  (T. 


(278.)  If  we  transfer  the  origin  of  co-ordinates  to  the  centre, 
M,  of  the  moon,  so  that  the  axis  of  z  shall  be  directed  toward 
the  pole,  the  axis  of  x  toward  a  point  whose  right  ascension  is 
a',  and  the  axis  of  y  toward  a  point  whose  right  ascension  is 
90°  -l-a',  these  co-ordinate  axes  will  be  parallel  with  those  before 
mentioned,  and  we  shall  have  for  the  co-ordinates  of  the  centre 
of  the  sun  referred  to  the  moon, 


ECLIPSES   OF    THE    SUN.  269 


a  a 

a  a 


parallel  to  the  new  axis  of  z,         Gr  sin.  D  ; 

?/,         Gr  cos.  D  sin.  (A  —  a') ; 
x,          G  cos.  D  cos.  (A— a'), 
since  the  right  ascension  of  the  sun's  centre,  seen  from  that  of 
the  moon,  is  A,  its  declination  is  D,  and  its  distance  is  Gr  (see 
page  264). 

Hence  we  have 

Gr  sin.  D  =  R  sin.  6/ — r  sin.  d, 
G-  cos.  D  sin.  (A— a')  =  —  r  cos.  6  sin.  (a— ax), 
Gr  cos.  D  cos.  (A — ax)=:R  cos.  6"  —  r  cos.  6  cos.  (a — a'). 

Dividing  each  equation  by  R,  and  putting  ^  —  g^  and  w=e, 

It  K 

we  shall  have 

g  sin.  D  =  sin.  6'— e  sin.  d,  } 

g  cos.  D  sin.  (A— 0.')=—  e  cos.  d  sin.  (a—  a/),  >  .  (6) 

g  cos.  D  cos.  (A— ax):=cos.  6'  —  e  cos.  6  cos.  (a— a7),  ) 
from  which  A,  D,  and  g  may  be  computed.     Dividing  the  sec- 
ond of  these  equations  by  the  third,  we  obtain 

/A        /v  e  cos.  6  sin.  (ct— a') 

tang.  (A— a')— ^ 1 

cos.  6'  —  e  cos.  6  cos.  (a— a7) 

e  cos.  6  sec.  6'  sin.  (a— ax) 
1  —  e  cos.  (5  sec.  <5'  cos.  (a— a')* 
Dividing  the  first  equation  by  the  third,  we  obtain 
T)_(sm-  &  —  e  sin.  6)  cos.  (A— of) 
cos.  d'  —  e  cos.  d  cos.  (a  —  ax)  ' 
Also,  from  equation  third, 

_cos.  cf  —  e  cos.  (5  cos.  (a  —  ax) 

cos.  D  cos.  (A— a') 

In  solar  eclipses  the  value  of  A  —  o!  never  exceeds  a  few  sec- 
onds, and  its  cosine  differs  from  unity  by  a  fraction  which  is  in- 
appreciable in  the  first  seven  decimal  figures ;  and  therefore  the 
factor  cos.  (A  — ax),  in  the  last  two  formulas,  may  be  suppressed. 
(279.)  The  preceding  expression  for  the  value  of  D  may  be 
converted  into  an  expression  for  the  value  of  D— 6/  by  omitting 
the  factor  cos.  (a— a7),  which  in  a  solar  eclipse  differs  but  little 
from  unity. 

By  Trig.,  Art.  77,  we  have 

tana  /A     TH-  tang.  A -tang.  B 
tang.  (A-B)= 


270 


PRACTICAL   ASTRONOMY. 


Hence 


tang.(D-(Jx):= 


sin.  6'  —  e  sin.  6 
cos.  6/  —  e  cos.  6 


-tang.  6' 


^  .sin.  6'  tang.  6'  —  e  sin.  (5  tang.  6' 
cos.  c5x  —  e  cos.  6 


_       sin.  6'  —  e  sin.  d—  sin.  (Sx  +  e  cos.  d  tang.  6' 


cos.  (5X  —  e  cos.  d+  sin.  6/  tang,  dx—  e  sin.  6  tang.  d' 

—  e  sin.  (5  cos.  dx-f-e  cos.  6  sin.  (5X 
cos.2  6'—e  cos.  d  cos.  (5x  +  sin.2  6'  —  e  sin.  d  sin.  dx' 


That  is, 


But  since  d—  (5X  is  a  small  arc,  we  may,  without  material  error, 
substitute  the  arc  for  its  sine,  and  we  may  also  use  the  arc 
D-eT  instead  of  its  tangent,  and,  neglecting  the  factor  cos.  (d-d'), 
we  obtain 


1  —  e 


In  the  same  manner,  we  obtain 


A=a'- 


Also, 


e  cos.  6  sec.  6/  (a— of) 

\  —  e  cos.  (5*sec.  6' 
\  —  e  cos.  6  sec.  6' 


cos.  D  sec.  <5X 

or  g-  —  1  —  e,  very  nearly. 

(280.)  In  order  to  compute  #,  y,  etc.,  we  must  return  to  our 

original  system  of  co-ordinates,  page  264.     Conceive  about  the 

point  E  a  sphere  to  be  described  with  any  radius  at  pleasure, 

and  let  M  represent  the  moon's  place 
upon  this  sphere.  Let  P  represent 
the  pole  of  the  equator,  and  let  Z,  Y, 
X  represent  the  points  where  this 
sphere  is  intersected  by  the  positive 
ends  of  the  above-mentioned  axes. 
In  this  system,  the  point  P  will  lie 
in  the  plane  of  the  great  circle  ZY ; 
and  the  points  M,  Z,  Y,  and  X  will 
be  determined  respectively  by  the 
right  ascensions  and  declinations  a 
and  d,  A  and  D,  A  and  90° +  D, 

90°  -f  A  and  0°. 


ECLIPSES   OF  THE    SUN.  271 

The  co-ordinates  z,  y,  and  x  of  the  point  M,  in  respect  to  E, 
taken  parallel  to  the  ahove-mentioned  axes,  are  equal  to  the  pro- 
jections of  the  line  EM  =  r  on  these  axes,  or  to  the  products  of 
the  line  EM,  by  the  cosines  of  the  arcs  ZM,  YM,  and  XM.     The 
cosines  of  these  arcs  may  be  derived  from  the  spherical  triangles 
ZPM,  YPM,  and  XPM,  in  which  the  side  ZP=90°-D,  MP 
=  90°  -  d,  YP  =  D,  XP  =  90°  ;  also,  the  angle  ZPM  =  A  -  a,  YPM 
=  180°-(A-a),  and  XPM  =  90°  + A -a. 
Hence,  by  Spherical  Trigonometry,  Art.  225,  we  obtain 
z—r  [sin.  D  sin.  d+cos.  D  cos.  6  cos.  (a— A)],  ) 
y—r  [cos.  D  sin.  6— sin.  D  cos.  d  cos.  (a— A)],  ?...(?) 
x=r  cos.  d  sin.  (a— A). 

The  above  expression  for  the  value  of  y  is  subject  to  the  in- 
convenience of  furnishing  y  by  means  of  the  difference  of  two 
large  numbers.  We  may,  however,  easily  transform  it  into  an- 
other which  is  free  from  this  inconvenience. 

Since  cos.  x  =  cos.2  Jz  — sin.2  £x ;  that  is,  cos.  (a— A)  = 
cos.2  i(a— A)  —  sin.2  J(a— A),  and  sin.2  re-f  cos.2  a;=l,  by  substi- 
tution and  reduction  we  obtain 

z  =  r  [cos.  (d-D)  cos.2  j  (a- A) -cos.  (d+D)  sin.2  i(a-A)], 
y=r  [sin.  (3-D)  cos.2  i(a-A)+sin.  (d+D)  sin.2  J(a-A)j. 
(281.)  Having  thus  computed  z,  y,  and  x9  we  can  find  z',  y\ 
and  x'  by  the  following  expressions  : 

z'  =  z+(jr,  y'—y,  and  x'—x. 

Conceive  now  that  M  in  the  preceding  figure  no  longer  repre- 
sents the  moon's  centre,  but  the  geocentric  zenith  of  the  observ- 
er ;  the  declination  of  the  point  M  will  then  be  equal  to  <//,  or  the 
geocentric  latitude  of  the  place  of  observation ;  and  its  right  as- 
cension will  be  equal  to  p,  the  sidereal  time  of  the  observer  ex- 
pressed in  degrees.  If,  then,  we  represent  the  distance  of  the 
observer  from  the  centre  of  the  earth  by  p,  we  shall  obtain  the 
values  of  £,  77,  and  |  from  equations  (7),  by  substituting  p,  pt  and 
$'  in  place  of  r,  a,  and  6.  "We  thus  obtain 

£=p  [sin.  D  sin.  0'  +  cos.  D  cos.  0'  cos.  (p— A)],  ) 
q=p  [cos.  D  sin.  0'— sin.  D  cos.  $'  cos.  (p— A)],  ?  .  .  (8) 
£=p  cos.  <//  sin.  (JJL— A). 

(282.)  The  unit  to  which  the  lengths  of  the  lines  r,  R,  and  p 
are  referred  is  entirely  arbitrary.  Bessel  has  chosen  for  this  unit, 
as  being  most  convenient  for  computation,  the  equatorial  radius 


272  PRACTICAL   ASTRONOMY. 

of  the  earth.  If  we  represent  the  moon's  equatorial  horizontal 
parallax  by  TT,  the  sun's  mean  horizontal  parallax  by  TT',  and  the 
distance  from  the  centre  of  the  earth  to  that  of  the  sun  by  r', 
expressed  as  in  the  solar  tables,  where  the  mean  distance  of  the 
earth  from  the  sun  is  considered  as  unity ;  then,  if  the  equato- 
rial radius  of  the  earth  be  taken  as  unity, 

1  r' 

and  R  = 


sin.  TTX 

Let  H  represent  the  mean  radius  of  the  sun,  or  the  apparent 
radius  of  the  sun's  disk  at  the  distance  r'  =  1 ;  then  the  linear 
radius  of  the  sun,  or  k',  the  equatorial  radius  of  the  earth  being 
taken  as  unity,  will  be  represented  by 

sin.  H 


Consequently,  for  all  eclipses  of  the  sun  we  shall  have 

1  sin.  TTX  Gr .  sin.  TTX  1 

r= ;  e  — ;  g= 

sin.  TT          r'  sm.  TT  r' 

sin.  H  ±  k  sin.  Tr7 


sin.  /= 


(10) 


s .  tang.  /=  l  —  z  tang.  f±  k  sec.  /, 
where  the  sign  -f  applies  to  an  external  contact,  and  —  to  an  in- 
ternal contact. 

(283.)  The  numerator  of  the  expression  for  sin. /is  constant 
for  all  eclipses  of  the  sun.  From  the  transits  of  Venus  in  the 
years  1761  and  1769,  Encke  has  determined  7r/  =  8//.5776 ;  from 
Bessel's  measurements  at  the  transit  of  Mercury  in  the  year  1832, 
H  was  determined  —  959/x.788  ;  and  according  to  Burckhardt's 
tables  of  the  moon,  if  we  take  the  equatorial  radius  of  the  earth 
as  unity,  the  linear  radius  of  the  moon,  or  k,  will  be  equal  to 
0.2725.  Hence  we  have  generally 

log.  sin.  7^=5.6189407;  ) 

log.  (sin.  H-f  k  sin.  7?')  =  7.6688050 ;[  .  .  .  (11) 
log.  (sin.  R-k  sin.  T/)  =  7.6666896.  ) 

(284.)  Let  0  represent  the  geographical  latitude  of  a  given 
place,  <//  its  geocentric  latitude,  and  G>  the  east  longitude  of  the 
place  from  the  meridian  of  the  ephemeris  expressed  in  time. 

The  beginning  and  end  of  the  eclipse  can  nowhere  happen 
many  hours  before  or  after  the  middle  of  the  eclipse,  as  given  in 


ECLIPSES   OF    THE    SUN.  273 

the  ephemeris.  Let,  then,  T  represent  the  mean  solar  time  cor- 
responding to  the  middle  of  the  eclipse  under  the  meridian  for 
which  the  ephemeris  is  computed  ;  T  +  o>  will  be  the  correspond- 
ing mean  time  of  the  middle  under  the  meridian  of  the  given 
place.  If  we  represent  the  mean  time  of  the  beginning  or  end 
of  the  eclipse  at  the  given  place  by  T  +  w  + 1,  we  may  be  sure 
that  t  is  a  short  interval  of  time.  If  we  have  not  the  use  of  an 
astronomical  ephemeris,  we  may  employ  the  solar  and  lunar  ta- 
bles, and  may  assume  for  T  either  the  time  of  true  conjunction, 
or,  still  better,  the  time  of  middle  of  the  eclipse  for  the  earth 
generally. 

For  the  mean  times  T  — lh.,  T,  and  T  +  lh.,  compute  from 
the  ephemeris  the  values  of  a,  6,  and  TT  for  the  moon ;  also  a7,  <J', 
and  r'  for  the  sun.  Compute  from  equations  (6)  the  values  of 
A,  D,  and  g  ;  and  from  equations  (7)  the  values  of  z,  y,  and  x. 
Also,  compute  the  values  of  /  and  log.  i  from  equations  (4)  and 
(10).  Since  the  values  of  /  and  log.  i  change  but  slowly,  when 
only  an  approximate  computation  is  required,  we  may  assume 
that  these  quantities  remain  constant  throughout  the  entire  du- 
ration of  the  eclipse. 

(285.)  We  will  now  assume  that  for  the  mean  times  T  — lh. 
and  T  +  lh.,  under  the  meridian  of  the  ephemeris,  the  co-ordi- 
nates x,  T/,  and  z  have  the  values 

P—P',  q  —  q.',  b  —  b',  and  p+p',  q+q'  b  +  b'; 
which  values  will  be  general  for  all  parts  of  the  earth.  But  for 
the  given  place  we  must  also  compute  the  sidereal  times  which, 
under  the  meridian  of  the  place,  correspond  to  the  instants  when 
the  mean  times  T  —  lh.  and  T  +  lh.  occurred  under  the  meridian 
of  the  ephemeris.  We  then  compute  from  equations  (8)  the 
values  of  the  co-ordinates  |,  77,  and  £  for  the  given  place  ;  and  we 
will  assume  that  these  co-ordinates  for  the  two  instants  above 
mentioned  are 

u—u',  v  —  v',  w  —  w',  and  u-\-u',  v  +  v',  w+iv'. 

We  may  now  assume  approximately  that  at  the  time  T,  un- 
der the  meridian  of  the  ephemeris,  the  values  of  x,  y,  z,  |,  77,  and 
£  are  equal  to  p,  q,  b,  u,  v,  and  w,  and  that  the  hourly  varia- 
tions of  these  values  are  represented  by  ]/,  q',  b',  u',  v',  and  iv' '; 
also  that,  during  a  moderate  interval  of  time,  the  change  of  the 
preceding  values  is  proportional  to  the  time.  We  shall  there- 

S 


274  PRACTICAL   ASTRONOMY. 

fore  find  approximately  for  the  mean  time  T  -f  t,  under  the  me- 
ridian of  the  ephemeris, 

;  y  =  q+q't;  z  = 


where  t  is  expressed  in  hours  and  fractions  of  an  hour. 
Substituting  these  values  in  equation  (5),  we  obtain 


(286.)  In  order  to  facilitate  the  computation,  we  will  assume 
p—u—m  sin.  M  ;  p'—u'—n  sin.  N,  J 
q—v  —  m  cos.  M;  q'  —  v^n  cos.  N,  >    ...  (12) 

J-t£=L,  ) 

where  m  and  n  are  always  to  be  considered  positive.     Substi- 
tuting these  values,  we  obtain 

(m  sin.  M.+nt  sin.  N)2  +  (m  cos.  'M.  +  nt  cos.  N)2  =  L2 
By  expanding  this  equation,  we  obtain 

m2  sin.2  M  -\-2mnt  sin.  M  sin.  N  +  n2^2  sin.2  N 


_T  2 

cos.2  M+2w?^  cos.  M  cos.  N  +  ^2Z2  cos.2  N  )          ' 


But  since  sin.2  +  cos.2  =  l,  we  have 

m2+Zmnt  cos.  (M 
or 

m2  sin.2  (M  —  N)  +  m2  cos.2  (M  —  N)  +  2m  nt  cos.  (M  —  N)  +  nH2  =  L2; 
that  is,  m2  sin.2  (M  -  N)  +  [m  cos.  (M  -  N)  +  ?^]2  =  L2. 

Let  us  assume 

m  sin.  (M  —  N) 

-j;-     ~  =  sm.  V  .......  (13) 

then,  if  an  eclipse  actually  takes  place,  it  will  always  be  possi- 
ble to  compute  the  angle  o/>.  Substituting  sin.  ^  in  the  last  equa- 
tion but  one,  we  have 

L2  sin.2  ^>+[m  cos.  (M-N)-f^]2  =  L2, 
or        [m  cos.  (M—  N)+^]2  =  L2(1  —  sin.2  i/O-L2  cos.2  1/>. 
Extracting  the  square  root, 

m  cos.  (M—  ~N)-\-nt=  ±L  cos.  i/>, 
_w  cos.  (M  —  N)  +  L  cos.  i/) 

^  w 

where  the  unit  to  which  t  refers  is  the  mean  solar  hour. 

It  is  obvious  that  the  greater  of  the  two  values  of  t,  under- 
stood in  a  positive  sense,  must  correspond  to  the  end  of  the 
eclipse,  and  the  least  of  the  two  to  the  beginning.  Assuming 


ECLIPSES    OF   THE    SUN.  275 

the  angle  ^  to  be  taken  in  either  the  first  or  fourth  quadrant, 
we  find  for  the  given  place,  in  mean  time  of  the  place, 

Beginning  of  the  eclipse  at     T  +  w  --  cos.  (M  —  N)  --  cos.  i/>, 

Tl  'Yl 

End  "  "  T  +  w--  cos.  (M-N)  +  -  cos.  </>, 

n  n 

where  we  may  employ  the  value  of  /  instead  of  L  without  ma- 
terial error. 

(287.)  The  hourly  variations  of  |  and  77  may  be  found  by  dif- 
ferentiating the  values  of  £  and  77  in  equations  (8).  We  thus 
obtain 

d%  ..d(p—A) 

-±=p  cos.  <//  cos.  fr-A)  ^    ;, 

•ffi=-P  sin-  ¥  sin-  Vjy-p  cos.  <j>'  cos.  D  cos.  (M-A)— 

,/    •      T\    •      /         »v46i-rrA) 
+p  cos.  </>  sin.  D  sin.  (/x—  A)  u          , 

o>  J- 

or 


The  unit  of  time  is  here  taken  at  one  hour,  and  the  above 
values  must  be  expressed  in  parts  of  radius. 

(288.)  In  order  to  determine  on  what  point  of  the  sun's  disk 
the  first  and  last  contacts  will  take  place,  conceive  a  line  which 
passes  through  the  place  of  the  observer,  parallel  to  the  line 
which  joins  the  centres  of  the  sun  and  moon,  and  directed  to- 
ward the  positive  side  of  the  axis  of  z  ;  the  plane  which  passes 
through  this  line,  and  is  parallel  to  the  axis  of  y  (see  figure,  page 
263),  makes,  with  the  plane  which  passes  through  the  former 
line  and  the  apparent  place  of  the  moon,  the  angle  KON,  whose 

tangent  is  j^=  --  -.     Represent  this  angle  by  Q,.     Since  the 

sun  is  at  a  great  distance  from  the  earth  and  moon,  the  line 
which  joins  the  centres  of  the  sun  and  moon  at  the  time  of  an 
eclipse  forms  a  very  small  angle  with  that  which  passes  through 
the  place  of  the  observer  and  the  sun.  We  may  therefore  as- 
sume that  the  angle  Q  is  the  same  as  that  which  is  formed  at 
the  sun's  centre  by  the  hour  circle  of  the  sun,  and  that  circle 
which  passes  through  the  sun's  centre  and  the  point  of  the  sun's 


276  PRACTICAL   ASTRONOMY. 

limb  where  the  first  or  last  contact  takes  place.     Consequently, 
we  shall  have 

tang.  Q=±ll. 

y-n 

But  x—^—p  —  u-\-(p/  —  u/)t 

—  m  sin.  M.  +  nt  sin.  ft,  by  equation  12. 
Also,  nt=—m  cos.  (M  —  N)q=L  cos.  if). 

Hence 

x—^—m  sin.  M  —  m  sin.  N  cos.  (M—  N)q=L  cos.  i/>  sin.  N. 
But 
sin  M—  sin.  N  cos.  (M  —  N)  = 

=sin.  M—  sin.  N  cos.  M  cos.  N  —  sin.  N  sin.  M  sin.  N 
=sin.  M(l  —  sin.2  N)  —  sin.  N  cos.  M  cos.  N 
=  sin.  M  cos.  N  cos.  N—  sin.  N  cos.  M  cos.  N 
=cos.  N  sin.  (M  —  N). 
That  is, 

x—%=m  sin.  (M—  N)  cos.  N  +  L  cos.  o/>  sin.  IN". 
But  L=m8Jn.  (M-N) 

sin.  i/j 
Hence 

w  sin.  (M—  N)  cos.  N  sin.  i/>+w  cos.  -0  sin.  N  sin.  (M  —  N) 

X  -  C  -  -  ;  -  —  -  —  — 

sm.  i/; 

m  sin.  (M  —  N)  (          AT    .  .     AT 

=  —  —  -  ]  cos.  rs  sm.  W;=FSin.  N  cos. 

sin. 


=  +     sn.       +-. 
In  the  same  manner,  we  find 

y  —  77==FL  cos.  (Nq=V>). 
Hence,  for  the  first  contact, 

£-£:=:  -L  sin.  (N-V)  =  L  sin.  (N+180°-i/;), 
T/—  T/=—  L  cos.  (N—  i/>)  =  L  cos.  (N+1800  —  1/>); 
and  for  the  last  contact, 

x—  |=L  sin. 
y—f]  —  \±  cos. 
That  is,  for  the  first  contact, 

tang.  Q=tang.  (N  +  180°-i/>), 
or  Q^N+ISO0-^; 

and  for  the  last  contact, 

tang.  Q  =  tang. 
or 


ECLIPSES    OF   THE    SUN.  277 

(289.)  The  angle  Q,  is  measured  on  the  sun's  limb,  from  his 
north  point  by  the  east,  from  0°  to  360°.  If  we  conceive  an 
hour  circle  drawn  through  the  sun's  centre,  Q,  will  represent  the 
angle  comprehended  between  the  intersection  of  this  hour  circle 
with  the  sun's  disk  and  the  point  of  first  or  last  contact.  If  the 
observer  has  an  equatorial  telescope,  he  may  easily  determine 
the  north  point  of  the  sun's  disk  by  the  method  explained  in  Art. 
42.  If  the  telescope  is  not  equatorially  mounted,  we  must  refer 
the  points  of  first  and  last  contact  to  the  vertex  of  the  sun's  disk  ; 
for  which  purpose  we  must  compute  the  angle  P,  which  is  form- 
ed at  the  sun's  centre  by  an  hour  circle  and  a  vertical  circle,  as 
explained  in  Art.  145.  The  north  point  of  the  sun's  disk  will  be 
situated  to  the  right  of  the  vertex  if  the  sun  is  west  of  the  me- 
ridian, but  on  the  left  of  the  vertex  if  it  is  east  of  the  meridian. 

(290.)  The  following  is  a  recapitulation  of  the  formulae  em- 
ployed in  this  computation  : 

Let  T  represent  a  convenient  assumed  time  near  to  the  time 
of  conjunction.     Take  from  the  ephemeris,  for  two  or  three  full 
hours  preceding  and  following  T,  the  following  quantities  : 
a=:the  moon's  right  ascension,       of  —  the  sun's  right  ascension, 
<5=the  moon's  declination,  d'^the  sun's  declination, 

TT—  the  moon's  equ.  hor.  parallax,  r'^the  earth's  radius  vector. 

Then  compute  the  following  quantities  : 
sin.8x/.5776 


_ 


r'  sin.  TT 
log.  sin.  8/x.5776  =  5.6189407, 


.         .     e  cos.  d  sec.  d'ia—a') 

A   —  f*'  ___  _  _  \  _  f_ 

\  —  e  cos.  6  sec.  6/ 


l  —  e 

1  —  e  cos.  d  sec.  6/ 
cos.  D  sec.  <5'     ' 
x~r  cos.  6  sin.  (oc  —  A), 

y=r  sin.  (d-D)  cos.2i  (a-A)+r  sin.  (<J+D)  sin.2i  (a-  A), 
z=r  cos.  (6—  D)  cos.2i  (a—  A)-r  cos.  (d+D)  sin.2^  (a—  A), 

.      ,    7.6688050,,  ,. 

sm./=  ---  -  —   -  (for  an  external  contact), 

7.6666896  ,,  ,, 

sm./=  —  ~  (for  an  internal  contact), 


278  PRACTICAL    ASTRONOMY. 

t=tang./, 
k  =  0.2725, 

l=z  tang,  f+k  sec./. 

Compute,  also,  the  following  quantities  for  the  given  place, 
where 

p  —  the  sidereal  time  of  the  place  of  observation  ; 
//  =  the  same  for  the  meridian  of  the  ephemeris  ; 
6)  =  the  longitude  of  the  place  of  observation  ;  east  longitudes 

being  considered  positive,  west  longitudes  negative  ; 
(f)/=the  geocentric  latitude  of  the  place  of  observation  ; 
p—  the  earth's  radius  for  the  place  of  observation. 
£=p  cos.  0'  sin.  (IJL—  A), 

r]=  p  sin.  $'  cos.  D—  p  cos.  <//  sin.  D  cos.  (p—  A), 
$=p  sin.  $'  sin.  D-f  p  cos.  ft  cos.  D  cos.  (/u—  A), 
d%=  p  cos.  0X  cos.  (IJL—  A)  d(fj,—A), 
di]—^  sin.  D  d(p—  A)  —  £c?D, 
m  sin,  M  =  x—  |, 
w  cos.  M=y—  ?y, 
w  sin.  N^rc''—  c?^, 
w  cos.  'N=y/  —  dr], 

o/=rthe  hourly  variation  of  #, 
^x  =  the  hourly  variation  of  y. 
m  and  n  are  always  positive. 


iMj 

sin.  V=  T~  sm-  (M—  N). 
JL 

i/;  must  be  taken  in  the  first  or  fourth  quadrant. 

For  beginning  of  eclipse,   t1=  --  cos.  (M  —  N)  --  cos.  i/>. 

For  end  of  eclipse,  t^=  --  cos.  (M  —  N)+—  cos.  ib 

n  n 


Time  of  beginning  of  eclipse,  = 

Time  of  end  of  eclipse,  =  T  +  G>  -f  t,. 

Angle  from  north  point  for  beginning,  =180°-}-N—  0  =  Q,. 

Angle  from  north  point  for  end,  =  N  +  1[>  =  Q2. 

Angle  from  vertex,  nrQ-f-P^V. 

(291.)  Example.  It  is  required  to  compute  the  time  of  begin- 
ning and  end  of  the  solar  eclipse  of  July  28,  1851,  for  Cambridge 


ECLIPSES    OF    THE    SUN. 


279 


Observatory,  latitude  42°  22'  48"  north,  longitude  4h.  44m.  30s. 
west  of  Greenwich. 

The  right  ascension  and  decimation  of  the  moon  are  computed 
for  the  Nautical  Almanac  for  each  noon  and  midnight,  exam- 
ined hy  means  of  differences  to  the  fourth  order,  and  interpolated 
for  every  hour.  The  following  places  of  the  moon  for  several 
hours  before  and  after  conjunction  have  been  interpolated  from 
the  computed  places  in  the  Nautical  Almanac,  regard  being  had 
to  differences  of  the  fifth  order.  The  places  of  the  sun  have  also 
been  carefully  interpolated. 

For  the  Moon. 


Greenwich  mean  Solar 
Time. 

a  =  R.  A. 

<5=Dec. 

;r  =  Parallax. 

July  28,  0 

1 

"         2 
"         3 
4 
5 

125  40     6.75 
126  19     9'.41 

126  58  10.80 
127  37  10.82 
128  16     9.37 
128  55     6.36 

20     3  30.00 
19  58     9.36 
19  52  39.98 
19  47     1.91 
19  41  15.20 
19  35  19.88 

60  27.600 
60  28.710 
60  29.794 
60  30.851 
60  31.880 
60  32.882 

For  the  Sun. 


Greenwich  mean 
Solar  Time. 

a'  =  R.  A.                   (5'  =  Dec. 

Log.  r-  =  Log. 
Distance. 

/*'=  Greenwich  Sider. 
Time,  reduced  to  Arc. 

July  28,  0 
"        1 
2 
"        3 
4 
«        5 

127    6    5.25 
127     8  32.63 
127  10  59.99 
127  13  27.34 
127  15  54.67 
127  18  21.99 

19  5  24.70 
4  50.23 
4  15.74 
3  41.21 
3    6.64 
2  32.04 

0.0065782 
65761 
65739 
65718 
65697 
65675 

125  33  19.05 
140  35  46.90 
155  38  14.74 
170  40  42.59 
185  43  10.44 
200  45  38.29 

280 


PRACTICAL   ASTRONOMY. 


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ECLIPSES   OF   THE    SUN. 


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ECLIPSP:S    OF    THE    SUN. 


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ECLIPSES    OF  THE    SUN. 


285 


A  portion  of  the  labor  of  the  preceding  computation  may  be 
saved  by  the  use  of  a  table  by  Zech,  which  furnishes  the  loga- 
rithm of  the  sum,  or  difference  of  two  numbers  which  are  known 
only  by  their  logarithms.  This  table  is  contained  in  Hulsse's 
Sammlung  mathematischer  Tafeln.  Leipzig,  1849. 

The  following  are  the  results  for  #,  ?/,  and  z : 


Hour. 

X 

x' 

Diff. 

y 

* 

Diff. 

Log.  z 

Diff. 

0 

-1.338789 

+  .968508 

1.7546447 

+  569487 

-.083013 

421 

1 

-0.769302 

+  57 

+  .885495 

1.7546026 

+  .569544 

-.083379 

-366 

832 

2 

-0.199758 

j 

+  .802116 

1.7545194 

+  .569543 

-.083730 

-351 

1242 

3 

+0.369785 

-  60 

+  .718386 

1.7543952 

+  .569483 

-.084073 

-343 

1657 

4 

+0.939268 

-118 

+  63431? 

1.7542295 

+  .569365 

-.03441!: 

-340 

2068 

5 

+  1.508633 

+  .54990( 

1.7540227 

(292.)  The  preceding  quantities  are  independent  of  geograph- 
ical position,  and  serve  not  only  for  calculating  the  times  of  be- 
ginning, etc.,  of  the  eclipse  for  any  place  at  which  it  may  be 
visible,  but  also  for  the  calculations  requisite  to  determine  the 
longitude  of  a  place  from  the  observed  time  of  beginning  and  end. 

Computation  of  the  beginning  and  end  of  the  Eclipse  for 

Cambridge,  by  Formulce,  page  278. 
w=-71°  7/  30";  0'=42°  11'  21".l;  log.  ,9  =  9.9993429. 
For  a  first  approximation  we  will  assume  T  =  2h.  Greenwich 

p  cos.  0'  =  9.869121 
sin.  D  =  9.514161 
cos.  (p— A)  =  9.866437 
.17771  =  9.249719 


f/=     155°  38'  15" 

u=-  71°    7'  30" 

=       84°  30'  45" 

A=     127°  \V    2" 


jlt_A=-  42°  40'  17" 

p= 9.999343 
cos.  0'  =  9.869778 
sin.  fc- A)  =  9.831097ft 
f==~.50144=9.700218» 

p= 9.999343 

sin.  0'  =  9.827098 

9°  4X  9X/  cos.  =  9.975489 

.63377  =  9.801930 


7i  =+.45606 

p  sin.  <//  =  9.826441 

sin.  D  =  9.514161 

.21908  =  9.340602 

pcos.  0X  =  9.869121 

cos.  D  =  9.975489 

cos,  (fi-  A)  =  9.866437 

.51410  =  9.711047 

^=+.73318 


286 


PRACTICAL   ASTRONOMY. 


The  hourly  variation  of  p  —  •  A, 
that  is, 

d(fi-A.)  =  15°  0'5".6, 
which,  in  parts  of  radius,  is 
.2618265. 

pcos.  <j/  =  9.869121 
cos.  (p-  A)  =  9.866437 
d(p-  A)  =  9.418014 
df=+.  14242  =  9.153572 
log.  £=  9.700218ft 
sin.  D  =  9.514161 
%-  A)  =  9.418014 
-.04289  =  8.632393ft 
dD=-33".8, 
which,  in  parts  of  radius,  is 
.0001638. 

log.  £=9.8652 

dD=  6.2143» 
-.00012  =  6.0795ft 
£fy=-.  04277 

£=-.19976 
|=-.  50144 
x  -£=+.30168 
y=  .80212 
T-  .45606 


y-ri=     .34606 

log.  (x-  1)  =  9.479546 

log.  (y-  n]  =  9.539151 

tang.  M  =  9.940395 

M  =  41°4/50// 
log.  (z-  1)  =  9.479546 
sin.  M  =  9.817644 


^=+.56954 
<%=+.!  4242 
tf  -(%=+.  42712 
^=-.  08355 
^=-.04277 
y>-dri=-.  04078 
log.  (^-^)  =  9.630550 
log.  (T/X  -  ^)  =  8.610447^ 
tang.  N  =  1.020103ft 
N  =  95°  27X  14/x 


log.  (o/-^)  = 

sin.  N  =  9.998029 
n  =  9.632521 

log.  £=9.8652 
1=7.6632 

^=.00338  =  7.5284 
/=.  53418 
L  =  .  53080 

M=  41°    4X  50/x 
N=   95°  27X  14/x 
M-N  =  305°  37  36/x 
sin.  (M-N)  =  9.910000» 

m  =  9.661902 
L  comp.  =  0.275069 
sin.  v>  =  9.846971ft 
i/;  =  315°19/47// 

cos.  (M-N)  =  9.765297 
m  =  9.661902 
comp.  ^  =  0.367479 
-f.  62327  =  9.794678 

cos.  ^  =  9-851970 
L  =  9.724931 
comp.  rc  =  0.367479 
+  .87979  =  9.944380 


ECLIPSES    OF    THE    SUN. 


287 


Hence          tl=  -1.50306h. ;  fa= +0.25652h. 

Beginning =  T  +  ^=:0.49694h.  Greenwich  time, 

End.  . =  T  +  *a  =  2.25652h. 

which  are  only  to  be  considered  as  approximate  values. 

For  a  second  approximation  we  will  assume  0.5h.  for  the  be- 
ginning, and  2.25h.  for  the  end. 


Beginning. 

End. 

T 

0.5h.  Gr.  m.  t. 

2.25h.  Gr.  m.  t. 

0         /         /. 

0 

V-' 

133  4  32.97 

159  23  51.7 

G) 

-  71  7  30. 

-  71  7  30. 

jU  —  //  +  W 

61  57  2.97 

88  16  21.7 

A 

127  7  28.36 

127  11  37.34 

ttr-A 

-  65  10  25.39 

-  38  55  15.64 

D 

19  4  59.66 

19  4  0.50 

p  cos.  (// 

9.8691208 

9.8691208 

sin.  Qi—  A) 

9.9578873^ 

9.7981314^ 

log.  £ 

9.8270081^ 

9.6672522ft 

f 

-.671441 

-.464785 

p  sin.  0' 

9.8264412 

9.8264412 

cos.  D 

9.9754523 

9.9754953 

log.  p  sin.  <//  cos.  D 

9.8018935 

9.8019365 

p  cos.  <// 

9.8691208 

9.8691208 

sin.  D 

9.5144700 

9.5141097 

cos.  (/*  —  A) 

9.6231132 

9.8909867 

log.  p  cos.  0'  sin.  D  cos.  (fi—  A) 

9.0067040 

9.2742172 

p  sin.  0'  cos.  D 

.633714 

.633777 

p  cos.  0'  sin.  D  cos.  (p—  A) 

.101556 

.188026 

S 

+  .532158 

+  .445751 

p  sin.  0X 

9.8264412 

9.8264412 

sin.  D 

9.5144700 

9.5141097 

log.  p  sin.  0'  sin.  D 

9.3409112 

9.3405509 

p  cos.  <fS 

9.8691208 

9.8691208 

cos.  D 

9.9754523 

9.9754953 

cos.  (p—  A) 

9.6231132 

9.8909867 

log.  p  cos.  0'  cos.  D  cos.  (fi—  A) 

9.4676863 

9.7356028 

p  sin.  <//  sin.  D 

.219236 

.219054 

p  cos.  <//  cos.  D  cos.  (ft  —  A) 

.293553 

.544005 

£ 

+  .512789 

+  .763059 

p  cos.  0' 

9.8691208 

9.8691208 

cos.  (/z  —  A) 

9.6231132 

9.8909867 

6?(^-A) 

9.4180136 

9.4180136 

log.  rf| 

8.9102476 

9.1781211 

*? 

+  .081329 

+  .150703 

288 


PRACTICAL   ASTRONOMY. 


Beginning.                              End. 

log.  I 

9.8270081ft         9.6672522ft 

sin.  D 

9.5144700           9.5141097 

d([L—  A) 

9.4180136           9.4180136 

log.  £  sin.  D£%—  A) 

8.7594917ftl         8.5993755ft 

log.  « 

9.7099 

9.8826 

dD 

6.2143ft 

6.2143ft 

log.  ^D 

5.9242ft 

6.0969ft 

|  sin.  Df/(^  —  A) 

-   .057477 

-  .039754 

$dD 

-   .000084 

-   .000125 

dr\ 

-   .057393 

-  .039629 

X 

-1.054056 

-0.057375 

f 

-0.671441 

-0.464785 

x—% 

-0.382615 

+  0.407410 

y 

+  0.927049 

+  0.781216 

n 

+  0.532158 

+  0.445751 

y-n 

+  0.394891 

+  0.335465 

log.  (x-t) 

9.5827620ft         9.6100317 

log.  (y  —  rj) 

9.5964772           9.5256472 

tang.  M 

9.9862848ft 

0.0843845 

M 

315°  54X  16/x.4 

50°  31"  54X/.0 

log.  (a—!) 

9.5827620ft 

9.6100317 

sin.  M 

9.8425192w         9.8876038 

m 

9.7402428           9.7224279 

a/ 

+  0.569487 

+  0.569543 

« 

+  0.081329 

+  0.150703 

a/  —  c?| 

+  0.488158 

+  0.418840 

y' 

-0.083013 

-0.083643 

drj 

-0.057393 

-0.039629 

y'—drj 

-0.025620 

-0.044014 

log.  (x'  —  dg) 

9.6885604 

9.6220482 

log.  (f/'-cty) 

8.4085791ft 

8.6435908ft 

tang.  N 

1.2799813ft 

0.9784574ft 

M 

93°  Ox  15x/.5 

95°  59X  56/x.2 

log.  (x'  —  dZ;} 

9.6885604 

9.6220482 

sin.  N 

9.9994027 

9.9976152 

^ 

9.6891577 

9.6244330 

log.  < 

9.70994 

9.88256 

i* 

7.66324 

7.66325 

log.  /^ 

7.37318 

7.54581 

ft 

002361 

.003514 

/ 

534242 

.534162 

L=/—  ^ 

531881 

.530648 

M-N 

222°  547  0/x.9 

314°  31X  57/x.8 

ECLIPSES    OF   THE    SUN. 


289 


Beginning. 


End. 


sin.  (M-N) 

9.8329711^ 

9.8529982ft 

m 

9.7402428 

9.7224279 

comp.  L 

0.2741855 

0.2751934 

sin.  i/> 

9.8473994^ 

9.8506195ft 

¥ 

315°  16'  25x/.8 

314°  50X  59/x.8 

cos.  (M-N) 

9.8648313ft 

9.8459141 

m 

9.7402428 

9.7224279 

comp.  ft 

0.3108423 

0.3755670 

9.9159164ft 

9.9439090 

-.823979 

+  .878838 

COS.  l/> 

9.8515508 

9.8483445 

L 

9.7258145 

9.7248066 

comp.  ft 

0.3108423 

0.3755670 

9.8882076 

9.9487181 

+  .773050 

+  .888624 

* 

+  .050929 

+  .009786 

Beginning  of  Eclipse. 

End  of  Eclipse 

T  +  £ 

0.550929h., 

2.259786h., 

or 

or 

Greenwich  mean  time 

Oh.  33m.  3.3s. 

2h.  15m.  35.2s. 

60 

4h.  44m.  30s., 

4h.  44m.  30s., 

or 

or 

Cambridge  mean  time 

7h.  48m.  33.3s. 

9h.  31m.  5.2s. 

N 

93°     Ox 

96°    Ox 

g 

315°  16X 

314°  51X 

ft 

180°  +  N—  1/> 

N  +  T/> 

=  317°  44' 

=  50°  51X 

These  results  agree  well  with  those  found  on  page  253. 

(293.)  We  may  obtain  a  check  upon  the  accuracy  of  our  com- 
putations in  the  following  manner  : 
Equation  (5),  page  267,  is 


all  the  quantities  being  supposed  to  be  computed  for  the  instant 
of  first  or  last  contact  of  the  limbs  of  the  sun  and  moon.  If 
these  quantities  have  been  computed  for  a  time,  T,  which  differs 
from  the  instant  of  contact  by  a  small  interval,  £,  they  may  be 
reduced  to  the  instant  of  contact  by  means  of  the  quantities  x', 
y',  d%,  and  cfy,  which  represent  the  hourly  variations  of  x,  y,  £, 
and  r.  In  this  case  we  shall  have 


290  PRACTICAL   ASTRONOMY. 

Thus,  in  the  preceding  example,  for  the  beginning  of  the 
eclipse,  #=+.050929. 

x-$=  -0.382615 
x'-dtt=  +  0.024861 


Sum  =-0.357754,  whose  square  is  .127988. 
Also,  y—n=  +0.394891 

'-drt  =  -0.001305 


Sum=  +0.393586,  whose  square  is  .154910. 
The  sum  of  these  two  squares  is  .282898,  which  is  the  square 
of  .531881,  the  value  of  L. 

For  the  end  of  the  eclipse,  t=  +.009786. 

a-f=  +0.407410 
(x'-d£)t=+  0.004099 

Sum=  +0.411509,  whose  square  is  .169340. 
Also,  #-77=  +0.335465 

(y'-dri)t=-  0.000431 

Sum  =     0.335034,  whose  square  is  .112248. 
The  sum  of  these  two  squares  is  .281588,  which  is  the  square 
of  .530648,  the  value  of  L. 

(294.)  "When  the  highest  accuracy  is  not  required,  the  labor 
of  the  preceding  computations  may  be  diminished  by  substitu- 
ting approximate  formulse  for  some  of  those  here  used.  The 
expressions  for  A,  D,  and  g*,  given  on  page  277,  may  be  simpli- 
fied without  greatly  diminishing  their  accuracy.  Since  e  is  al- 
ways a  small  quantity,  the  denominators  of  the  expressions  for 
A  and  D  are  nearly  equal  to  unity,  and  may  be  omitted.  More- 
over, at  the  time  of  an  eclipse,  6,  6',  and  D  are  very  nearly  equal 
to  each  other  ;  hence  the  following  expressions  will  afford  a  good 
approximation  to  the  values  of  A,  D,  and  g*. 

A  =  a'-e(a-a'), 
T>=d'-e(d-6')9 
g=l-e. 

'  These  formulae  will  furnish  the  values  of  A  and  D  within  a 
small  fraction  of  a  second. 

For  the  remaining  computations  we  must  proceed  according 
to  the  formulae  on  pages  277-8. 


ECLIPSES   OF   THE    SUN.  291 

SECTION   Y. 
BESSEL'S  METHOD  OF  COMPUTING  OCCULTATIONS  OF  STARS. 

(295.)  The  formulas  required  for  the  computation  of  occulta- 
tions  of  stars  by  the  moon  are  easily  deduced  from  those  already 
given  for  solar  eclipses,  since  the  distance  of  the  fixed  stars  is 
such  that  they  have  no  diurnal  parallax,  and  the  rays  of  light 
which  emanate  from  them  and  touch  the  moon's  disk  may  be 
considered  as  forming  the  surface  of  a  cylinder.  Hence,  for  oc- 
cultations,  the  quantities  f  and  i9  as  well  as  the  horizontal  par- 
allax of  the  star,  become  each  equal  to  zero  ;  also,  a/  =  A.J  6/  =  Dy 
and  l=k.  It  is  unnecessary  to  compute  either  z  or  £.  Since  A, 
the  right  ascension  of  the  star,  is  invariable,  c?(^—  A)  becomes  dp. 
But  the  variation  of  p  in  one  solar  hour  is  Ih.  Om.  9.8565s.,  or 
15°  2/  27/x.85,  which,  in  parts  of  radius,  is  .2625162,  whose  log- 
arithm  is  9.4191561.  For  a  solar  eclipse,  the  angle  Q,  was  re- 
ferred to  the  sun's  limb,  but  in  an  occultation  of  a  star  this  an- 
gle is  referred  to  the  moon's  limb,  and  in  the  latter  case  the  an- 
gle Q,  will  differ  180°  from  the  angle  Q,  in  the  former  case. 
Hence  we  have  the  following  formulae  for  the  computation  of 
occultations  : 

T  =  any  convenient  assumed  time  near  conjunction  ; 

a  =  the  moon's  right  ascension  ; 

(J=the  moon's  declination; 

7T=the  moon's  equatorial  horizontal  parallax  ; 

A=the  star's  right  ascension  ; 

D^the  star's  declination  ; 

p,  =  thQ  sidereal  time  of  the  place  of  observation  ; 
//—  the  same  for  the  meridian  of  the  ephemeris  ; 

w—  the  longitude  of  the  place  of  observation  ;  east  longi- 

tudes positive,  west  longitudes  negative  ; 
§'  —  the  geocentric  latitude  of  the  place  of  observation  ; 

p^the  earth's  radius  for  the  place  of  observation  ; 
cos.  6  sin.  (a  —  A) 

r    -  V  '    • 


_sin.  (d-D)  cos.2  ^(a-A)  +  sin.  (d+D)  sin.2  j(a-A) 

sin.  TT 

|=p  cos.  0'  sin.  (p—  A)  ; 
ri—p  sin,  <//  cos.  D—  p  cos.  </>'  sin.  D  cos.  (^—  A)  ; 

—    cos.    '  cos. 


292 


PRACTICAL   ASTRONOMY. 


dr\  =  %  sin.  ~Ddfj, ; 
log.  «^= 9.4191561; 
m  sin.  M=a;— |; 
in  cos.  M=^ — 77 ; 
^  sin.  N  =  o;'— c?|; 
w  cos.  N=yx — o??/ ; 

re' = the  hourly  variation  ofx; 
I/  =  the  hourly  variation  of  y  ; 
m  and  n  are  always  positive ; 

sin.  ^  —  -r  sin.  (M  — N) ; 

-0  must  be  taken  in  the  first  or  fourth  quadrant ; 
log.  k =9.4353665; 

m          .,,-    ,TV     k 

^= cos.  (M  — N) —  cos.  i/> ; 

n  n 

t«= cos.  (M  —  N)+—  cos.  -0. 

n  n 

Time  of  immersion  =T  +  a>-f£i. 
Time  of  emersion    =  T  +  w  +  £2. 
For  immersion,  angle  from  north  point  toward  east, 

For  emersion,  angle  from  north  point  toward  east, 


Angle  from  vertex  =  Y= 

(296.)  Ex.  1.  Required  the  time  of  occultation  of  a  Tauri, 
January  23,  1850,  for  Cambridge  Observatory. 

We  find  that  the  apparent  conjunction  takes  place  at  about 
13  hours  Greenwich  mean  time.  We  therefore  interpolate,  from 
the  computed  places  in  the  Nautical  Almanac,  the  moon's  places 
for  several  hours  before  and  after  conjunction,  regard  being  had 
to  differences  of  the  fifth  order,  as  on  page  279,  and  obtain  the 
following  results : 


Greenwich  mean  Time. 

a. 

"$. 

7T. 

h. 
11 

12 
13 
14 
15 

65  47  '  3.70 
66  24     3.30 
67     1     7.95 
67  38  17.62 
68  15  32.29 

+  16  23  28.91 
16  30  4.10 
16  36  33.22 
16  42  56.21 
16  49  12.97 

59  48.046 
59  50.000 
59  51.942 
59  53.871 

59  55.785 

The  position  of  a  Tauri  is 

A =66°  497  53x/.l;  D  =  -f  16°  127  3/x.4. 


OCCULTATIONS     OF     STARS. 


293 


•i 


COOS  0*rH  O  GO 

GO  O  O  CD  O 

JS  O  *O  rH  GO  CD 

OJ  rH  GO  GO  »O  CO 

,  t>  CD  Q5  t>  rH 

I    CO  ^  O5  rH  O 


CO1OCOCOO}T3H<H/CD 

~    -~   rH  CD  CO  CO 


TF  rH 


~w"CDCDO}CDlOi>CDCD(MCD»O*OJ>*O 
O5O5O^*OCDCOGOlO^COCDaOO5GO 
^  ^OSOiGOOGOOJOSlOlOCDOOlOOS 

Oi  Ot)  CO  ^O  L^  I?**  ^xt^  ^«O  C^  L^*  t^  ^O 

0CO  OO5OicO*TfrHO5CO"'^G6oirHt> 


O 

CO 


CD 


CD  O 

GO  CD  § 
00  t>  O 
GO  GO  rt 


rf  CD  Oi  J>  00  J> 

CO  ^  Oi  rH  O5   i 


^CD  GO  OJ  OJ  GO  O  CD 

^  ^  O5  Oi  GO  O  GO 

"  rH  1> 

'lO  rv-)  ^^  W3  'ij  '-JU  u«5  r-l  «J  'JU  TT  C.^  »O  OD  rH  O  rH 
OiO5CvJCDJ>t>rHCDGOJ>i>rHlOO'O 
0O5  OO5O5CO'^tlrHO5CO'<sJ*I>O5rHl> 
CO 


I 


2    aoiOO^ojcocD 

I>  t>  CO  O  CO  CO 

rH  Oi  CD  CO 
IQ  05  00  00  O 


CO 


m  1>  CD  TH 

Oi  rH  O5  | 


O  GO  GO 


O)  rH  O  O  *^  CO  t*» 

00  CD  CO  CD  O  Tt<  Tj* 
^T  Oi  O5  rH  r— I  O  rH 


°co       o  o  co  ^  r-  o 


o  TH  CD 


294 


PRACTICAL   ASTRONOMY. 


The  following  are  the  results : 


Hour. 

X 

af 

Diff. 

y 

y'    \  Diff. 

11 

-1.007841 

+  .193633 

+  .593905 

+  .107845 

12 

-  .413936 

+80 

+  .301478 

-24 

+  .593985 

+  .107821 

13 

+  .180049 

+  16 

+  .409299 

-34 

+  .594001 

+  .107787 

14 

+  .774050 

-44 

+  .517086 

-48 

+  .593957 

+  .107739 

15 

+  1.368007 

+  .624825 

For  the  first  trial,  we  may  assume  T  =  13  hours  Greenwich 
mean  time,  and  we  shall  obtain  the  approximate  times  of  immer- 
sion and  emersion.  As,  however,  this  example  has  already  been 
computed  on  page  259,  we  will  suppose  the  approximate  times 
to  be  known,  and  will  assume  12  hours  for  immersion,  and  13.25 
hours  for  emersion.  The  work  will  then  be  as  follows : 


Immersion. 

Emersion. 

T 

12h.  G-r.  m.  t. 

13.25h.  Gr.  m.  t 

X 

-.413936 

+  .328551 

y 

+  .301478 

+  .436250 

X' 

+  .593945 

+  .593997 

y' 

+  .107833 

+  .107795 

p-A 

-14°59/54//.67 

+  3°  48X  10".14 

p  COS.  (j)' 

9.8691208 

9.8691208 

sin.  (p—  A) 

9.4129543^ 

8.8216638 

log.* 

9.2820751^ 

8.6907846 

S  ; 

-.191459 

+  .049066 

p  sin.  fi 

9.8264412 

cos.  D 

9.9824020 

log-  (1) 

9.8088432 

(1) 

+  .643937 

+  .643937 

p  COS.  <$>' 

9.8691208 

9.8691208 

sin.  D 

9.4456150 

9.4456150 

cos.  (j»—  A) 

9.9849468 

9.9990427 

log.  (2) 

9.2996826 

9.3137785 

(2) 

+  .199380 

+  .205958 

(l)-(2)=, 

+  .444557 

+  .437979 

p  COS.  $' 

9.8691208 

9.8691208 

cos.  (p—  A) 

9.9849468 

9.9990427 

a? 

9.4191561 

9.4191561 

log.  <% 

9.2732237 

9.2873196 

k 

+  .187596 

+  .193785 

OCCULTATIONS     OF     STARS. 


295 


Immersion. 

Emersion. 

log.  1 

9.2820751ra 

8.6907846 

sin.  D 

9.4456150 

9.4456150 

d\i 

9.4191561 

9.4191561 

log.  dri 

8.1468462^ 

7.5555557 

dn 

-.014023 

+  .003594 

x-S 

-.222477 

+  .279485 

y-f] 

-.143079 

-.001729 

log.  (x-$) 

9.3472852^ 

9.4463585 

log.  (y-rj) 

9.1555759^ 

7.2377950^ 

log.  tang.  M 

0.1917093 

2.2085635^ 

M 

237°  15X  15/x.l 

90°  21'  16".0 

sin.  M 

9.9248366^ 

9.9999917 

log.  m 

9.4224486 

9.4463668 

x'-d$ 

+  .406349 

+  .400212 

y'-dr) 

+  .121856 

+  .104201 

log.  (^-^|) 

9.6088992 

9.6022901 

log.  (T/'-cty) 

9.0858469 

9.0178719 

log.  tang.  N 

0.5230523 

0.5844182 

N 

73°  187  25/x.4 

75°  247  22".4 

sin.  N 

9.9813010 

9.9857571 

log.  ^ 

9.6275982 

9.6165330 

M-N 

163°  56X  49x/.7 

14°  56X  53/x.6 

sin.  (M-N) 

9.4417330 

9.4115288 

log.  m 

9.4224486 

9.4463668 

comp.  & 

0.5646335 

0.5646335 

log.  sin.  i/> 

9.4288151 

9.4225291 

rf 

15°  34X  13X/.0 

15°  20X  27/x.5 

cos.  (M-N) 

9.9827265^ 

9.9850487 

log.  m 

9.4224486 

9.4463668 

comp.  w 

0.3724018 

0.3834670 

9.7775769^ 

9.8148825 

-.599207 

+  .652954 

COS.  1/J 

9.9837624 

9.9842430 

log.& 

9.4353665 

9.4353665 

comp.  n 

0.3724018 

0.3834670 

9.7915307 

9.8030765 

+  .618772 

+  .635443 

t 

-.019565 

-.017511 

=  -  70.4s. 

=  -  63.0s. 

Hence  we  have  the  following  results : 

Greenwich  mean  Time.  Cambridge  mean  Time. 

Time  of  immersion,    llh.  58m.  49.6s.,  or  7h.  14m.  19.6s. 

"       emersion,       13h.  13m.  57.0s.,  "  8h.  29m.  27.0s. 


296  PRACTICAL   ASTRONOMY. 

For  immersion,  Qi  —  N—  i/>  =   57°  44'. 

«    emersion,     Q2=180°  +  N  +  V  =  270°  45'. 
These  results  are  nearly  the  same  as  found  on  page  261. 
(297.)  We  may  obtain  a  check  upon  the  accuracy  of  our  com- 
putations in  the  same  manner  as  shown  for  a  solar  eclipse  on 
page  289.     Equation  (5),  page  267,  "becomes,  in  the  case  of  an 
occultation, 

(x  -  £)2  +  (y-  if  =  k2  =  .074256, 

the  quantities  x,  y,  |,  and  77  being  supposed  to  be  computed  for 
the  instant  of  immersion  or  emersion.  If  these  quantities  have 
been  computed  for  a  time,  T,  which  differs  from  the  instant  of 
immersion  or  emersion  by  a  small  interval,  £,  we  shall  have 


Thus  in  the  preceding  example,  for  immersion,  t  —  —  .019565. 

x  -|-=-.  222477 
(x/-d^)t=-.  007950 

Sum  =  -.230427,  whose  square  is  .053097. 
Also,  y—  77  =-.143079 

(y'-dr])t=-.  002384 

Sum  =  —.145463,  whose  square  is  .021159. 
The  sum  of  these  two  squares  is  .074256. 
For  emersion,  £=—.017511. 

a-!=+.  279485 
(x/-d£)t=-.  007008 

Sum=     .272477,  whose  square  is  .074244. 
Also,  y-  77  =-.001729 

'-drt=-.  001825 


Sum=     .003554,  whose  square  is  .000012. 

The  sum  of  these  two  squares  is  .074256. 

(298.)  Ex.  2.  Required  the  time  of  occultation  of  y  Yirginis, 
January  9,  1855,  for  Washington  Observatory. 

Apparent  conjunction  takes  place  between  23  and  24  hours 
of  Greenwich  mean  time.  The  moon's  places  for  23,  24,  and 
25  hours,  Greenwich  time,  according  to  the  American  Nautical 
Almanac,  are  as  follows  : 


Greenwich  m.  t. 

a. 

I 

7T. 

23 
24 
25 

12  33  21.97 
12  35  12.33 
12  37    2.80 

+  0    8  30.8 
-0     5  32.3 
-0  19  36.2 

55  31.18 
55  32.80 
55  34.43 

OCCULTATIONS     OF     STARS. 


297 


The  position  of  y  Virginis  is 

34m.  18.43s.;  D= 


-0°  39X 


Computation  of  the  co-ordinates  x  and  y. 


Greenwich  mean  Time. 

23h. 

24h. 

25h. 

a-A 

-14'  6".9 

+  13X  28/x.5 

+41X  5x/.55 

log.   7T 

3.5225981 

3.5228093 

3.5230216 

log.  sin.  —  log. 

4.6855560 

4.6855560 

4.6855559 

sin.  TT 

8.2081541 

8.2083653 

8.2085775 

log.  (a-A) 

2.9278321^ 

2.9076800 

3.3919138 

log.  sin.  —  log. 

4.6855736 

4.6855738 

4.6855645 

cos.  6 

9.9999987 

9.9999994 

9.9999929 

cosec.  TT 

1.7918459 

1.7916347 

1.7914225 

log.  x 

9.4052503/z 

9.3848879 

9.8688937 

X 

-.254244 

+  .242598 

+  .739424 

(5+D 

-30'  41".3 

_44'  447/.4 

-58'  48/x.3 

(5-D 

+47X  42/x.9 

+33X  39/x.8 

+  19X  35/x.9 

cos.  J(a—  A) 

9.9999991 

9.9999992 

9.9999922 

cos.  J(a  —  A) 

9.9999991 

9.9999992 

9.9999922 

log.  IP-  D) 

3.4568062 

3.3053084 

3.0703704 

log.  sin.  —  log. 

4.6855609 

4.6855679 

4.6855725 

cosec.  re 

1.7918459 

1.7916347 

1.7914225 

log.  (1) 

9.9342112 

9.7825094 

9.5473498 

log.  i(a-A) 

2.62680 

2.60665 

3.09088 

log.  sin.  —  log. 

4.68557 

4.68557 

4.68557 

sin.  i(a  —  A) 

7.31237 

7.29222 

7.77645 

sin.  (<5  +  D) 

7.95069^ 

8.10440w 

8.23250^ 

cosec.  TT 

1.79185 

1.79163 

1.79142 

log.  (2) 

4.36728ra 

4.48047^ 

5.57682^ 

(1) 

+  .859431 

+  .606051 

+  .352655 

(2) 

-.000002 

-.000003 

-.000038 

(i)+(2)=y 

+  .859429 

+  .606048 

+  .352617 

The  following  are  the  results  : 


Hour.      |             x 

of 

Diff. 

y 

y' 

Diff. 

23 
24 
25 

-.254244 
+  .242598 
+  .739424 

+  .496842 
+  .496826 

-16 

+  .859429 
+  .606048 
+  .352617 

-.253381 
-.253431 

-50 

For  a  first  approximation  we  assume  T  =  24  hours  Greenwich 
mean  time.  The  corresponding  sidereal  time  at  Washington  is 
14h.  9m.  35.06s.  ;  whence 


Also,  p  sin.  </>'=  9.7955439,  and  p  cos.  ^  9.8917226. 


298 


PRACTICAL   ASTRONOMY. 


Hence  we  obtain 

x=  +.24260;  |=  +.31474;  m  sin.  M  =  - .07214. 
#=  +  . 60605;  17  =+.63261;  m  cos.  M=-. 02656. 

M=249°  47"  16/x,  log.  m  =  8.885779. 

o/=  +  .49683;  cg=+.18716;  n  sin.  N  =  +  .30967. 

y'=  -.25341 ;  dq=  -.00094 ;  n  cos.  N=  -.25247. 

N  =  129°  11'  24",  log.  *=9.601567. 

V>  =  14°  3' 12". 
ti==- .56368 ;  t2  =  +  .75955. 

Greenwich  mean  Time.    Washington  mean  Time. 

Time  of  immersion  =  23.43632h.,  or  18h.  17m.  59s. 
"       emersion    =  24.75955h.,  "  19h.  37m.  23s. 
For  the  second  approximation  we  will  assume  23h.  25m.  for  im- 
mersion, and  24h.  45m.  for  emersion.    The  results  are  as  follows : 


Immersion. 

Emersion. 

T 

23h.  25m.  Gr.  m.  t.  24h.  45m.  GT.  m.  t. 

X 

-.047224 

+  .615218 

y 

+  .753860 

+  .415979 

X' 

+  .496843 

+  .496821 

y' 

-.253377 

-.253443 

^A 

15°  2X  43x/.2 

35°  6'  0/x.3 

f 

+  .202302 

+  .448121 

H 

+  .633058 

+  .631747 

ds        •  , 

+  .197574 

+  .167383 

dr\ 

-.000606 

-.001341 

M 

295°  49X  58x/.2 

142°  14X  41//.2 

log.  m 

9.4428399 

9.4360119 

N 

130°  llx  6'y.3 

127°  25X  29x/.9 

log.  n 

9.5929879 

9.6178712 

* 

14°  367  247/.7 

14°  48X  44x/.4 

t 

+  .012470 

-.001011 

ft 

115°  35' 

322°  14X 

Hence  We  have  Greenwich  mean  Time.  Washington  mean  Time. 

Time  of  immersion,  23h.  25m.  44.9s.,  or  18h.  17m.  33.7s. 
"       emersion,     24h.  44rn.  56.4s.,  "  19h.  36m.  45.2s. 
Check.  —  For  immersion, 

x  -%=-.  249526 
(s/-<%X=+.  003732 

Sum  =—.245794,  whose  square  is  .060414. 
Also,  y-«?=  +  .120802 

=-.  003152 


Sum=  +.117650,  whose  square  is  .013842. 


OCCULTATIONS     OF     STARS.  $99 

The  sum  of  these  two  squares  is  .074256. 
For  emersion, 

z-£:=+.  167097 
(x'-dt)t=-.  000333 

Sum=  +.166764,  whose  square  is  .027810. 
Also,  y-n=  -.215768 

y'-d^t  =+  .000255 


Sum=  —  .215513,  whose  square  is  .046446. 

The  sum  of  these  two  squares  is  .074256. 

In  the  Tables  from  which  the  American  Nautical  Almanac  is 
computed,  the  value  of  k  is  assumed  to  be  0.272278.  In  the 
English  Nautical  Almanac  for  1857  the  value  of  k  is  assumed 
to  be  0.273114.  The  value  employed  in  Burckhardt's  Tables 
of  the  Moon  is  0.2725. 

(299.)  In  the  American  Nautical  Almanac,  and  also  in  the 
Berlin  Jahrbuch,  are  furnished  elements  by  which  the  preceding 
computations  are  materially  abridged.  These  elements  are  the 
co-ordinates  x  and  #,  with  their  hourly  variations.  In  the  Amer- 
ican Almanac,  p.  375-397,  is  given  a  list  of  all  the  stars,  to  the 
sixth  magnitude  inclusive,  contained  in  the  B.  A.  Catalogue, 
which  can  be  occulted  by  the  moon.  It  also  furnishes  for  each 
star  the  Washington  mean  time  (T)  of  conjunction  with  the 
moon  ;  the  Washington  hour  angle  of  the  star  at  the  time  T  ; 
and  the  co-ordinates  for  the  same  time,  with  their  hourly  varia- 
tions. At  the  instant  of  conjunction  x  reduces  to  zero,  and  is 
therefore  omitted  from  the  almanac. 

Thus  for  y  Yirginis,  January  9,  1855,  we  find  on  page  375, 
T=  Washington  mean  time  of  conjunction,  18h.  22.5m. 

H  —  Washington  hour  angle  of  star  at  time  T,  +  lh.  5m.  51s. 
Y=the  co-ordinate  which  Bessel  represents  by  #,  +0.7298 
pf—  hourly  variation  of  p,  Bessel  represents  by  x',  +0.4968 
q'  =  hourly  variation  of  #,  Bessel  represents  by  y',  —0.2530 
log.  sin.  D  =  log.  sine  of  star's  declination,  —  8.0570 

log.  cos.  D^log.  cosine  of  star's  declination,  0.0000 

Having  the  assistance  of  these  numbers,  we  are  relieved  from 
the  necessity  of  the  preliminary  computations  on  page  297,  and 
the  approximate  times  of  immersion  and  emersion  are  obtained 
with  very  little  labor,  especially  if  we  employ  logarithms  to  only 
four  decimal  places,  which  will  generally  furnish  results  correct 
to  the  nearest  minute. 


Ax 


CHAPTER  XII. 

LONGITUDE. 
SECTION   I. 

LONGITUDE    DETERMINED    BY    TRANSPORTATION    OF    CHRONOMETERS. 

(300.)  THE  manufacture  of  chronometers  has  attained  to  such 
a  degree  of  perfection  as  to  afford  the  means  of  determining  the 
difference  of  longitude  of  two  stations,  not  too  remote  from  each 
other,  with  a  precision  superior  to  that  of  most  other  methods. 
The  following  are  the  essential  steps  of  this  method :  The  time 
is  accurately  determined  at  one  station,  Greenwich,  for  instance, 
and  the  chronometer  is  carefully  compared  with  the  transit  clock ; 
hence  the  error  of  the  chronometer  on  the  meridian  of  Green- 
wich is  known.  The  chronometer  heing  carried  to  a  second  sta- 
tion, for  example,  Cambridge  Observatory,  is  compared  with  the 
transit  clock  at  that  place.  Thus  the  error  of  the  chronometer 
on  the  meridian  of  Cambridge  is  known ;  but  its  error  on  the 
meridian  of  Greenwich  at  the  same  instant  is  known,  if  its  rate 
be  known,  and  the  longitude  is  the  difference  of  these  two  er- 
rors. In  grand  chronometric  expeditions,  it  is  customary  to  em- 
ploy a  large  number  of  chronometers,  from  twenty  to  fifty,  or 
more,  as  checks  upon  each  other. 

(301.)  The  most  serious  difficulty  in  the  application  of  this 
method  consists  in  determining  the  rate  of  the  chronometers 
during  the  journey,  for  chronometers  generally  have  a  different 
rate  when  transported  from  place  to  place,  either  by  land  or  by 
sea,  from  that  which  they  maintain  in  an  observatory.  When 
it  is  proposed  to  determine  the  difference  of  longitude  of  two  sta- 
tions with  the  greatest  accuracy,  the  error  of  the  chronometers 
should  be  determined  at  the  commencement  of  the  expedition, 
at  the  first  station ;  the  same  thing  should  be  done  at  the  sec- 
ond station ;  then,  as  soon  as  possible,  the  chronometers  should 
be  brought  back  to  the  first  station,  and  their  error  determined 
anew.  The  chronometers  should  thus  be  transported  back  and 
forth  a  considerable  number  of  times. 


LONGITUDE.  301 

Let  us  designate  the  eastern  station  by  A,  the  western  by  B, 
and  the  west  longitude  of  the  place  B  from  A  we  will  designate 
by  G).  We  will  suppose  that  at  the  time  £,  at  the  place  A,  the 
error  of  one  of  the  chronometers  was  a  ;  that  on  its  arrival  at  B, 
at  the  time  t',  the  error  was  b  ;  and  again,  on  its  return  to  A  at 
the  time  £/x,  the  error  was  a'.  If  we  regard  a  day  as  the  unit 
of  time,  and  represent  the  mean  daily  rate  of  the  chronometer 
during  the  journey  by  w,  we  shall  have 

_  of  —  a  t 

whence  we  may  conclude  that 

(*)  =  a  -fw  (t'—t)  —by 
or  w  =  a' — m(t" —t')  —  b. 

Each  chronometer  will  afford  an  independent  determination 
of  the  value  of  w ;  and  in  order  to  detect  any  irregularity  in  the 
rates  of  the  chronometers,  they  should  be  compared  daily  with 
each  other  throughout  the  entire  journey. 

The  following  observations  were  made  to  determine  the  differ- 
ence of  longitude  between  two  stations,  A  and  B : 


Station  A 
Station  B 
Station  A 


t  —  September  15,  11.55h. 
*'==  September  17,  18.85H. 
t"  =  September  18,  11.55h. 


a=+ 34m.  20.1s. 
b=+  31m.  0.6s. 
a'=+34m.  4.4s. 


1  5  7s 
Consequently  we  have  m=—      '      =  —  5.23s. 

o 


.  20.1s. 

m(t'-t)=-  5.23  x  2.304  =  -  12.1s. 

-b=  -31m.    06s. 

Longitude  =  «—       3m.    7.4s. 

(302.)  Since  chronometers  almost  invariably  indicate  a  differ- 
ent rate,  according  as  they  are  traveling  or  at  rest,  if  the  observer 
remains  for  several  days  at  the  station  B,  the  error  of  the  chro- 
nometers should  be  determined  immediately  upon  arrival,  and 
again  before  departing  from  B  ;  and  the  interval  of  rest  should 
not  be  included  in  the  determination  of  the  value  of  m.  Sup- 
pose we  have  determined  the  chronometer  errors 

o,  &,  £',  a', 
corresponding  to  the  times 

/  i'  //x  ///x 
t,  i  ,  i  ,  i    , 

where  a  and  a'  are  supposed  to  have  been  obtained  at  the  place 


302 


PRACTICAL   ASTRONOMY. 


A  ;  b  was  the  error  on  first  arriving  at  B,  and  b'  the  error  on  de- 
parting from  B. 

Then  the  interval  of  time  embraced  in  the  two  journeys  is 


and  the  change  in  the  error  of  the  chronometer  for  the  same  time 
is  of—  <£  —  &  —  b. 


Hence  we  have     ™ 


The  following  example  is  taken  from  Struve's  chronometric 
expedition,  undertaken  in  1843,  between  Pulkova  and  Altona  : 


Pulkova 
Altona 
Altona 
Pulkova 
Here  we  have 


*=May  19,  21.54h. 
tf  ==  May  24,  22.66h. 
*"  =  May  26,  10.72h. 
31,    O.OOh. 


a=          +  6m.  38.10s. 

b=-lh.  14m.  39.92s. 

£x=:-lh.  14m.  36.77s. 


a'= 


+  7m.    9.58s. 


Also, 


Hence 


a'  —  a  =  31.  48s. 
b'-b  =  3.15s. 
(of  -  a)  -(b/-b)=  28.33s. 

r-t=5d.    1.12h. 
*'"-*"  =  4d.  13.28h. 
-t")  =  9d.  14.40H. 
28.33s. 


m  — 


9.6 


-=+  2.951s. 


a=          +6m.  38.10s. 

m(V  -t)= 2.951  x  5.047  =  +  14.89s. 

-b=  +  lh.  14m.  39.92s. 

Longitude  =  w—     Ih.  21m.  32.91s. 

(303.)  It  is  here  assumed  that  the  rate  of  the  chronometer  was 
the  same  during  the  journey  from  Pulkova  to  Altona  as  during 
the  journey  from  Altona  to  Pulkova.  In  order  to  eliminate  any 
error  which  might  arise  from  this  supposition,  Struve  begins  the 
next  calculation  with  Altona,  so  that  any  change  in  the  rate  of 
the  chronometer  will  produce  the  opposite  effect  from  that  which 
would  result  if  the  computation  commenced  with  Pulkova.  The 
following  combination  is  the  one  which  immediately  succeeded 
that  of  the  former  example : 


Altona 
Pulkova 
Pulkova 
Altona 


*=May  26,  10.78k 
^  =  May31,  O.OOh. 
t"  —  June  3,  5.62h. 
7,  20.52h. 


fc=-lh.  14m.  36.77s. 

a=  +7m.  9.58s. 
a'=  +  7m.  19.36s. 
^=-lh.  14m.  0.35s. 


LONGITUDE.  303 


Here  we  have  b'—b  =  36.42s. 

a'-a  =  9.78s. 


-a)  =  26.64s. 

f_f=4d.  13.28h. 
j'"_*":=4d.  14.90h. 

-*"  =  9d-    418h- 


6=-lh.  14m.  36.77s. 
4.553  =  +  13.22s. 

-a=         -7m.    9.58s. 
Longitude^  G>  =—  Ih.  21m.  33.13s. 

In  the  chronometric  expedition  already  referred  to,  nine  voy- 
ages were  made  from  Pulkova  to  Altona,  and  eight  from  Altona 
to  Pulkova,  in  which  81  chronometers  were  employed.  The  re- 
sults of  13  of  this  number,  having  shown  greater  discordances 
than  the  rest,  were  rejected,  and  the  deduced  longitude  was  made 
to  depend  upon  68  chronometers. 

This  result  was 

Ih.  21m.  32.52s., 
with  a  probable  error,  according  to  Struve,  of  only  0.04s. 

(304.)  It  is  indispensable  to  the  accuracy  of  these  results  that 
the  time  be  obtained  at  each  station  with  the  greatest  precision. 
Struve  recommends  that  the  time  be  determined  with  a  tran- 
sit instrument,  by  observations  of  stars  near  the  zenith,  inas- 
much as  a  slight  deviation  of  the  transit  instrument  from  the 
plane  of  the  meridian  does  not  affect  the  time  of  passage  of  a 
zenith  star.  It  is  necessary,  however,  to  know  the  inclination  of 
the  axis  with  the  greatest  accuracy  ;  and  the  axis  should  be  re- 
versed upon  its  supports  during  each  series  of  observations,  so  as 
to  eliminate  the  effect  of  unequal  pivots  and  of  collimation  error. 
In  order  to  eliminate  the  effect  of  any  error  in  the  right  ascensions 
of  the  stars  employed,  the  same1  stars  should,  if  possible,  be  ob- 
served at  both  stations.  For  this  purpose,  a  catalogue  of  all  the 
stars  which  pass  near  the  zenith,  and  of  a  magnitude  sufficient 
to  be  observed  without  inconvenience,  should  be  prepared  before- 
hand, and  a  copy  furnished  to  each  observer.  If  the  places  of 
any  of  the  stars  are  too  imperfectly  known,  they  should  be  care- 
fully observed  with  the  instruments  of  some  large  observatory. 


304  PRACTICAL    ASTRONOMY. 

(305.)  The  comparisons  of  the  chronometers  should  all  be 
made  by  observing  the  coincidence  of  beats.  If  we  undertake 
to  compare  two  clocks  which  beat  seconds  of  the  same  kind  of 
time,  unless  they  happen  to  tick  at  the  same  instant,  there  is  a 
fraction  of  a  second  which  must  be  estimated  by  the  ear.  This 
estimation  is  extremely  difficult,  and  practiced  observers  will 
differ  among  themselves  by  a  quarter  of  a  second,  and  sometimes 
even  more.  When,  however,  the  two  clocks  happen  to  tick  to- 
gether, there  is  no  fraction  of  a  second  to  be  estimated ;  and  a 
practiced  ear  will  detect  any  deviation  from  coincidence  in  beats 
amounting  to  0.01s.  Now  a  sidereal  clock  gains  upon  a  solar 
clock  one  second  in  about  six  minutes ;  and  if  two  such  clocks 
are  placed  side  by  side,  they  must  tick  together  once  in  every 
six  minutes.  In  order  to  compare  two  such  clocks,  we  notice 
their  movements,  and  wait  until  the  beats  sensibly  coincide, 
when  we  know  that  their  difference  amounts  to  an  entire  number 
of  seconds,  which  is  readily  discovered.  Chronometers  generally 
make  two  beats  in  a  second ;  so  that  between  a  clock  which 
beats  seconds  of  sidereal  time,  and  a  chronometer  which  ticks 
half  seconds  of  solar  time,  there  must  be  a  coincidence  every 
three  minutes.  Chronometers  are  sometimes  made  to  tick  13 
times  in  6  seconds.  Such  a  chronometer,  regulated  to  mean 
time,  makes  121  ticks  in  56  seconds  of  sidereal  time ;  that  is, 
the  coincidences  between  such  a  chronometer  and  a  sidereal  sec- 
onds-pendulum would  occur  every  56  seconds.  Moreover,  the 
intervals  between  the  ticks  of  the  chronometer  is  0.4628s.  side- 
real time  ;  and  13  of  these  intervals  are  equal  to  6.016s.  sidereal 
time  ;  54  are  equal  to  24.991s. ;  67  are  equal  to  31.007s. ;  and 
121  are  equal  to  55.999s ;  that  is,  in  the  course  of  56  seconds 
there  are  five  coincidences  within  the  limits  ±  0.02s.  Such  a 
chronometer  affords  the  means  of  comparing  by  coincidences 
with  great  rapidity;  a  consideration  of  no  trifling  importance 
where  80  chronometers  are  to  be  compared  daily.  Chronome- 
ters are  frequently  made  to  beat  five  times  in  two  seconds,  which 
gives  a  coincidence  at  every  36  seconds  with  a  half-second  side- 
real chronometer. 

(306.)  It  is  also  indispensable  to  the  accuracy  of  the  results 
that  the  personal  equation  of  all  the  observers  employed  in  ob- 
taining the  time  should  be  carefully  determined.  The  mode  of 


LONGITUDE.  305 

doing  this  has  already  been  explained  on  page  80.  This  correc- 
tion is  the  most  difficult  to  obtain  satisfactorily,  especially  as 
persona]  equation  is  not  always  a  constant  quantity,  but  is  liable 
to  vary  with  the  physical  condition  of  the  observer.  It  is  the 
opinion  of  Mr.  Airy,  that  when  a  tolerable  number  of  chronom- 
eters is  used  for  a  moderate  distance,  and  in  good  observing 
weather,  the  variation  of  personal  equation  is  the  error  to  be 
most  apprehended. 

A  grand  chronometric  expedition  has  been  for  several  years  in 
progress,  at  the  expense  of  the  United  States  coast  survey,  for 
the  purpose  of  determining  the  difference  of  longitude  between 
Greenwich  and  Cambridge,  Massachusetts.  A  large  number 
of  chronometers  have  been  transported  by  means  of  the  Cunard 
steamers  from  the  Liverpool  Observatory  to  Cambridge,  and  back 
again  to  Liverpool.  During  the  summer  of  1849,  forty-four 
different  chronometers  were  employed  in  several  trials,  and 
during  the  progress  of  the  expedition  more  than  four  hundred 
exchanges  of  chronometers  have  been  made.  For  facility  of 
comparing  the  chronometers,  Mr.  Bond  used  a  chronometer  beat- 
ing half  seconds,  and  gaining  12  minutes  daily  on  mean  solar 
time,  which  furnished  a  coincidence  of  beats  every  90  seconds. 


SECTION    II. 

LONGITUDE    DETERMINED    BY    THE    ELECTRIC    TELEGRAPH. 

(307.)  The  difference  of  the  local  times  of  two  places  may  be 
determined  by  means  of  any  signal  which  can  be  seen  or  heard 
at  both  places  at  the  same  instant.  When  the  places  are  not 
very  distant,  the  explosion  of  a  rocket  or  the  flash  of  gunpowder 
may  serve  this  purpose.  Six  or  eight  ounces  of  powder  at  night 
makes  a  good  signal  at  a  distance  of  twenty-five  to  thirty  miles  ; 
but  for  a  distance  of  ten  miles,  two  or  three  ounces  are  sufficient, 
if  the  observers  are  provided  with  telescopes. 

(308.)  But  the  electric  telegraph  affords  the  means  of  trans- 
mitting signals  to  a  distance  of  a  thousand  miles  or  more  with 
scarcely  any  appreciable  loss  of  time.  The  first  experiments  of 
this  kind  any  where  made  were  undertaken  in  the  United  States ; 
and,  with  the  exception  of  a  rude  experiment  of  Captain  Wilkes 
in  1844,  all  the  experiments  in  this  country  have  been  made  in 

U 


306  PRACTICAL   ASTRONOMY. 

connection  with  the  United  States  Coast  Survey.  Suppose  there 
are  two  observatories  at  a  considerable  distance  from  each  other, 
and  that  each  is  provided  with  a  good  clock  and  a  transit  instru- 
ment for  determining  its  error ;  then,  if  they  are  connected  by  a 
telegraph  wire,  they  have  the  means  of  transmitting  signals  at 
pleasure  from  either  observatory  to  the  other,  for  the  purpose  of 
comparing  their  local  times.  The  signal  is  given  at  either  sta- 
tion by  pressing  a  key,  as  in  the  usual  mode  of  telegraphing ; 
and  the  observer  at  the  other  station  hears  the  click  caused  by 
the  motion  of  the  armature  of  his  electro-magnet.  Four  dif- 
ferent methods  of  comparison  have  been  practiced  in  the  exper- 
iments by  the  United  States  Coast  Survey. 

(309.)  The  first  method  is  the  most  obvious  one,  and  consists  in 
simply  striking  on  the  signal  key  at  intervals  of  ten  seconds ;  the 
party  at  one  station  recording  the  time  when  the  signals  were 
given,  and  the  other  party  recording  the  time  when  the  signals 
were  received.  After  about  twenty  signals  have  been  transmit- 
ted from  the  first  station  to  the  second,  a  similar  set  of  signals 
is  returned  from  the  second  station  to  the  first.  This  mode  of 
comparison  has  but  one  serious  imperfection,  and  this  is,  that 
it  requires  the  fraction  of  a  second  to  be  estimated  by  the  ear. 
The  party  giving  the  signals  strikes  his  key  in  coincidence  with 
the  beats  of  his  clock,  so  that  at  this  station  there  is  no  fraction 
of  a  second  to  be  estimated ;  but  at  the  other  station  the  arma- 
ture click  will  not  probably  be  heard  in  coincidence  with  the 
beats  of  the  clock,  and  the  fraction  of  a  second  is  to  be  estimated 
by  the  ear.  Now  this  fraction  can  not  be  estimated  with  the 
accuracy  which  is  demanded  in  this  kind  of  comparison.  It  is 
found  that  observers  generally  estimate  the  fraction  of  a  second 
too  small  when  using  the  ear  alone,  unassisted  by  the  eye.  This 
error  is  greatest  at  the  middle  date  between  two  clock  beats,  and 
is  found  to  vary  from  0.06  to  0.18  of  a  second  with  different  ob- 
servers. 

(310.)  This  evil  suggested  the  second  method  of  observation, 
which  relies  on  the  coincidences  of  a  mean  solar  and  sidereal 
clock  or  chronometer.  The  following  is  the  method  pursued : 
After  transmitting  a  few  signals  by  the  former  method,  so  as  to 
determine  the  difference  between  the  local  times  of  the  two  sta- 
tions within  a  small  fraction  of  a  second,  the  party  at  the  first 


LONGITUDE. 


307 


station  commences  striking  on  his  signal  key  every  second,  in 
coincidence  with  the  beats  of  his  mean  solar  chronometer,  and 
continues  to  do  so  for  ten  or  fifteen  minutes  without  interruption. 
The  party  at  the  second  station  compares  the  armature  click  of 
his  magnet  with  the  beats  of  his  sidereal  clock,  and  watches  for 
a  coincidence,  and  records  the  time  when  the  coincidence  takes 
place.  When  he  has  obtained  two  or  three  coincidences,  which 
generally  requires  from  ten  to  fifteen  minutes,  he  breaks  the  elec- 
tric circuit,  in  order  to  notify  the  first  party  to  stop  beating.  He 
then  commences  beating  seconds  by  striking  his  own  signal  key 
in  coincidence  with  the  beats  of  his  sidereal  clock ;  and  the  party 
at  the  first  station  compares  the  armature  clicks  of  his  magnet 
with  the  beats  of  his  solar  chronometer,  and  watches  for  a  coin- 
cidence. When  he  has  obtained  three  or  four  coincidences, 
which  generally  requires  ten  or  twelve  minutes,  he  breaks  the 
electric  circuit,  in  order  to  notify  the  other  party  to  stop  beating. 
The  comparison  of  times  at  the  two  stations  is  now  complete. 

(311.)  The  following  observations  were  made  August  1,  1849, 
for  the  purpose  of  determining  the  difference  of  longitude  be- 
tween the  High  School  Observatory  in  Philadelphia  and  Western 
Reserve  College  Observatory  at  Hudson,  Ohio.  The  time-keeper 
employed  at  Philadelphia  was  a  mean  solar  chronometer,  beat- 
ing half  seconds ;  the  time-keeper  at  Hudson  was  a  sidereal 
clock. 


Signals  given  at  Philadelphia, 
mean  Time. 

Signals  received  at 
Hudson,  sid.  Time. 

Signals  given  at   |    Signals  received  at 
Hudson,  sid.  Time.  iPhiladel.,  mean  Time. 

h.      m.      s. 

h.       m.        s. 

h.      m.        s. 

h.       m.          s. 

14  15  30 

17  44  23.5 

18  13     0 

14  44     1.6 

40 

33.5 

10 

11.8 

50 

43.5 

20 

21.8 

16     0 

53.7 

30 

31.9 

10 

45     3.8 

40 

41.8 

20  20 

49  14.2 

50 

51.7 

30 

24.3 

14     0 

45     1.7 

40 

34.4 

10 

11.4 

50 

44.4 

20 

21.6 

21     0 

54.5 

30 

31.6 

10 

50     4.5 

40 

41.4 

Result,  14  20  40 

17  49  34.4 

18  14     0 

14  45     1.7 

From  these  comparisons  we  may  conclude  that  14h.  20m.  40s. 
on  the  Philadelphia  chronometer  corresponds  to  17h.  49m.  34.4s. 


308 


PRACTICAL   ASTRONOMY. 


on  the  Hudson  clock ;  and  ISh.  14m..  Os.  on  the  Hudson  clock 
corresponds  to  14h.  45m.  1.7s.  on  the  Philadelphia  chronometer. 
The  Philadelphia  observer  beat  seconds  for  ten  minutes,  and 
two  coincidences  were  recorded  at  Hudson  ;  .viz.,  at 
17h.  58m.    Os., 
18h.    4m.  10s. 

The  Hudson  observer  beat  seconds  for 'eleven  minutes,  and 
three  coincidences  were  recorded  at  Philadelphia  ;  viz.,  at 
14h.  54m.  25s., 
14h.  57m.  28.5s., 
15h.    Om.  39s. 
The  former  comparisons  show  us  that 

By  the  Philadelphia  Chronometer.  By  the  Hudson  Clock. 

14h.  29m.    4s.  correspond  to  17h.  58m.    Os. 

14h.  35m.  13s.  "  18h.    4m.  10s. 

14h.  54m.  25s.          "  18h.  23m.  25s. 

14h.  57m.  28.5s.        «  18h.  26m.  29s1. 

15h.    Om.39s.          «  18h.  29m.  40s. 

At  16h.  Om.  the  Philadelphia  chronometer  was  4h.  48m. 
25.78s.  fast,  and  losing  2.22s.  per  day. 

At  18h.  6m.  the  Hudson  clock  was  8.13s.  fast,  and  losing  1.02s. 
per  day. 

In  the  following  table,  column  first  shows  the  corrected  Phil- 
adelphia mean  times  ;  column  second  the  corresponding  Phila- 
delphia sidereal  times ;  column  third  the  corrected  Hudson  si- 
dereal times ;  and  column  fourth  shows  the  differences  between 
the  numbers  in  the  two  preceding  columns,  or  the  difference  of 
longitude  between  the  two  places. 


Philadel.  mean  Time. 

Philadelphia  sid.  Time. 

Hudson  sid.  Time. 

Diff.  of  Longitude. 

'9  40  38.08 
9  46  47.09 
10     5  59.12 
10     9     2.62 
10  12  13.13 

h.       m.          s. 

18  22  57.43 
18  29     7.45 
18  48  22.63 
18  51  26.64 
18  54  37.66 

h.      m.          s. 

17  57  51.87 
18    4     1.87 
18  23  16.89 
18  26  20.89 
18  29  31.89 

m.         s. 

25  5.56 

5.58 
5.74 
5.75 

5.77 

Mean  of  results  by  eastern  signals,    25rn.  5.57s. 
"          "          western  signals,  25m.  5.75s. 
Mean  of  both,  25m.  5.66s. 

The  difference  between  the  results  by  eastern  and  western  sig- 
nals is  partly  due  to  the  time  required  for  the  transmission  of  the 


LONGITUDE.  309 

signals ;  but  this  effect  disappears  from  the  mean  of  both  sets  of 
signals. 

(312.)  A  third  method  of  comparing  local  times  is  by  tele- 
graphing transits  of  stars.  This  method  was  practiced  in  the 
summer  of  1848,  between  New  York  and  Cambridge,  in  the  fol- 
lowing manner :  A  list  of  zenith  stars  is  selected  beforehand,  and 
furnished  to  each  observer.  When  every  thing  is  prepared  for 
observation,  the  Cambridge  astronomer  points  his  telescope  upon 
one  of  the  selected  stars  as  it  is  passing  his  meridian,  and  strikes 
the  key  of  his  register  at  the  instant  the  star  appears  to  coincide 
with  the  first  wire  of  his  transit.  He  makes  a  record  of  the 
time  by  his  own  chronometer,  and  the  New  York  astronomer, 
hearing  the  click  of  his  magnet,  records  the  time  by  his  own 
clock.  As  the  star  passes  over  the  second  wire  of  the  transit  in- 
strument, the  Cambridge  astronomer  again  strikes  the  key  of  his 
register,  and  the  time  is  recorded  both  at  Cambridge  and  New 
York.  The  same  operation  is  repeated  for  each  of  the  other 
wires.  The  Cambridge  astronomer  now  points  his  telescope 
upon  the  next  star  of  the  list,  which  culminates  after  an  inter- 
val of  five  or  six  minutes,  and  telegraphs  its  transit  in  the  same 
manner.  In  about  twelve  minutes  from  the  former  observation, 
the  first  star  passes  the  meridian  of  New  York,  when  the  New 
York  astronomer  points  his  transit  instrument  upon  the  same 
star,  and  strikes  the  key  of  his  register  at  the  instant  the  star 
passes  each  wire  of  his  transit.  The  times  are  recorded  both  at 
New  York  and  Cambridge.  The  second  star  is  telegraphed  in 
a  similar  manner.  The  same  operations  are  now  repeated  upon 
a  second  pair  of  stars,  and  so  on  as  long  as  may  be  thought  de- 
sirable. 

The  chief  objection  to  this  method  is,  that  it  involves  the  es- 
timation of  fractions  of  a  second,  as  in  the  usual  mode  of  transit 
observations ;  that  is,  it  involves  the  personal  equation  of  the 
observers. 

(313.)  The  fourth  method  of  comparison  obviates  this  evil  in 
some  degree,  by  printing  the  signals  upon  a  cylinder  or  a  fillet 
of  paper.  There  must  be  a  clock  at  one  of  the  stations  for 
breaking  the  electric  circuit  every  second,  as  described  in  Art. 
102 ;  and  there  must  be  a  register  at  each  of  the  stations  for 
recording  the  beats  of  the  clock  and  any  other  signals  which 


310  PRACTICAL   ASTRONOMY. 

may  be  required,  as  described  in  Art.  106.  When  the  connec- 
tions are  properly  made,  there  will  be  heard  a  click  of  the  mag- 
nets at  each  station  simultaneously  with  the  beats  of  the  electric 
clock,  and  the  registers  will  all  be  graduated  into  second  spaces. 
The  method  is  not  limited  to  two  stations,  but  any  number  of 
stations  may  be  compared  at  the  same  time.  In  January,  1849, 
Cambridge,  New  York,  Philadelphia,  and  Washington  were  con- 
nected in  this  manner.  The  mode  of  observation  is  the  same 
as  described  in  the  preceding  article,  except  that  the  observations 
are  all  recorded  by  the  operation  of  machinery.  The  Cambridge 
astronomer  strikes  the  key  of  his  register  as  the  star  passes  suc- 
cessively each  wire  of  his  transit  instrument,  and  the  dates  are 
printed  not  only  upon  his  own  register,  but  also  upon  those  at 
New  York,  Philadelphia,  and  Washington.  When  the  same  star 
comes  over  the  meridian  of  New  York,  the  observer  there  goes 
through  the  same  operation,  and  his  observations  are  printed 
upon  all  four  of  the  registers.  The  Philadelphia  observer  does 
the  same  when  the  star  comes  upon  his  own  meridian,  and  we 
proceed  in  the  same  manner  whatever  be  the  number  of  stations. 
Thus  we  have  four  or  more  registers  all  graduated  into  equal 
parts  by  the  ticking  of  the  same  clock,  and  upon  these  we  have 
printed  the  instants  at  which  the  star  was  seen  to  pass  each  wire 
of  the  transit  telescopes  at  the  several  stations.  These  observa- 
tions furnish  the  difference  of  longitude  of  the  stations,  inde- 
pendently of  the  tabular  place  of  the  star  employed,  and  also  in- 
dependently of  the  absolute  error  of  the  clock.  The  observers 
now  read  their  levels,  and  reverse  their  transit  instruments.  A 
second  star  is  now  telegraphed  successively  over  each  meridian, 
and  so  on  as  long  as  may  be  desired. 

(314.)  The  following  example  is  derived  from  observations 
made  in  the  summer  of  1852,  to  determine  the  difference  of 
longitude  between  Seaton  Station,  in  Washington,  District  of 
Columbia,  and  Roslyn  Station,  near  Petersburgh,  Virginia.  The 
observations  at  Seaton  were  recorded  upon  Bond's  spring  govern- 
or, and  those  at  Roslyn  upon  Saxton's  register,  and  also  a  Morse 
register.  The  diaphragms  of  the  transits  consisted  of  twenty- 
five  wires,  arranged  in  groups  of  five.  The  following  are  the  ob- 
servations of  star  6150,  British  Association  Catalogue,  July  7, 
1852,  with  the  complete  reduction  for  determining  the  clock  error : 


LONGITUDE. 


311 


Mean  of  all  the  wires, 

Reduction  to  middle  wire, 

Diurnal  aberration, 

Level  correction, 

Azimuth  correction, 

Collimation  correction, 

Personal  equation, 

Sum, 

Star's  right  ascension, 

Clock  error,  20.717 

The  following  table  shows  the  clock  errors, 
ilar  manner,  from  the  transits  of  15  stars,  on 
7th: 


Seaton  Station. 
h.     m.            s. 

18  2    9.341 

Roslyn  Station. 

18  3  45.108 

+  .030 

.000 

-.018 

-.018 

-.014 

-.076 

-.082 

-.156 

-.106 

-.192 

-.146 

-.000 

18  2     9.005 

18  3  44.666 

18  1  48.288 

18  1  48.288 

1  56.378 
derived  in  a  sim- 
the  night  of  July 


s,a, 

Clock  errors  at 

"  '  ! 
Difference. 

Seaton. 

Roslyn. 

" 

s 

m.          s. 

m.           s. 

6150,  B.  A.  C. 

20.717 

1  56.378 

1  35.661 

6268        « 

21.214 

56.710 

35.496 

6355        « 

20.977 

56.539 

35.562 

6404 

21.134 

56.705 

35.571 

6599        " 

20.912 

56.535 

35.623 

6667        " 

20.942 

56.663 

35.721 

6722 

21.023 

56.652 

35.629 

6784        " 

20.964 

56.659 

35.695 

7048        « 

21.398 

56.909 

35.591 

7114 

21.183 

56.768 

35.585 

7204        " 

21.017 

56.739 

35.722 

7277        « 

21.034 

56.713 

35.679 

7333        " 

21.015 

56.642 

35.627 

7398        " 

20.938 

56.347 

35.409 

7521 

20.917 

56.597 

35.680 

Mean  of  observations  July  7,  1852  1  35  617 

Mean  of  all  the  observations  on  six  nights   .  .     1  35.603 

(315.)  This  method  of  observation  is  so  accurate  as  to  furnish 
a  tolerable  measurement  of  the  velocity  of  the  electric  fluid.  If 
the  fluid  requires  no  time  for  its  transmission,  then  the  signals 
given  at  either  station  ought  to  be  similarly  printed  at  all  the 
stations ;  and  the  fraction  of  a  second  registered  upon  any  one 
scale  should  be  identically  the  same  as  upon  every  other.  But 
if  the  fluid  requires  time  for  its  transmission,  these  fractions  will 


312  PRACTICAL   ASTRONOMY. 

be  different.  Suppose  the  clock  to  be  at  "Washington ;  that  an 
arbitrary  signal  is  made  at  Cambridge ;  and  that  the  time  req- 
uisite for  the  transmission  of  a  signal  between  the  two  places  is 
the  thirtieth  of  a  second.  Then  the  clock-pause  will  be  regis- 
tered at  Cambridge  -g^th  of  a  second  after  it  took  place  and  was 
recorded  at  Washington,  and  the  arbitrary  signal-pause  will  be 
recorded  at  Cambridge  as  soon  as  it  is  made,  or  -^th  of  a  second 
before  it  reaches  Washington.  We  shall  thus  have  the  interval 
between  the  signal-pause  and  the  preceding  clock-pause  longer 
at  Washington  than  at  Cambridge,  and  the  excess  on  the  Wash- 
ington register  will  measure  twice  the  time  consumed  in  the 
transmission  of  the  signals  between  the  two  stations. 

Thus,  in  the  following  figure,  let  the  upper  line  represent  a 
portion  of  the  Washington  time  scale,  corresponding  to  15,  16, 

Washington,  » * » * * 

Cambridge, ^  -^ fl_  -&— 

etc.,  seconds,  and  the  lower  line  the  same  for  Cambridge,  each 
division  being  a  little  later  than  the  corresponding  one  for 
Washington.  Then,  if  an  arbitrary  signal  is  made  at  Cam- 
bridge between  16  and  17  seconds,  and  printed  at  A,  the  rec- 
ord on  the  Washington  scale  will  be  at  B,  and  the  interval  from 
16  to  B  will  exceed  that  from  16  to  A  by  twice  the  time  con- 
sumed in  the  transmission  of  the  signals  from  Cambridge  to 
Washington. 

Numerous  observations  have  been  made  under  the  direction 
of  the  superintendent  of  the  Coast  Survey  for  the  purpose  of  de- 
termining the  velocity  of  the  electric  fluid,  and  the  general  re- 
sult is  about  16,000  miles  a  second. 


SECTION    III. 

LONGITUDE    DETERMINED    BY    MOON-CULMINATING    STARS. 

(316.)  The  moon's  motion  in  right  ascension  is  very  rapid, 
amounting  to  about  one  minute  in  arc  for  every  two  minutes 
of  time. 

If,  then,  the  right  ascension  of  the  moon  has  been  observed  at 
two  different  stations,  we  may  infer  the  difference  of  longitude 
of  the  two  meridians  from  the  difference  of  the  observed  right 


LONGITUDE.  313 

ascensions  compared  with  the  times  of  observation.  If  we  have 
a  transit  instrument  adjusted  to  the  meridian,  and  observe  the 
passage  of  the  moon's  limb  and  some  known  star,  we  can  deduce 
the  right  ascension  of  the  moon's  limb  from  the  known  right  as- 
cension of  the  star.  If  we  select  for  comparison  a  star  which  is 
near  the  moon,  the  errors  of  the  instrument  will  have  but  little 
influence  upon  the  result,  since  these  errors  will  be  nearly  the 
same  for  the  moon  and  star.  The  English  and  American  Nau- 
tical Almanacs  both  furnish  the  moon's  place  and  those  of*  cer- 
tain neighboring  stars  on  every  day  upon  which  it  is  possible  to 
observe  the  moon.  These  stars  are  called  moon-culminating 
stars,  and  are  generally  four  in  number  for  each  day,  two  pre- 
ceding and  two  following  the  moon,  and  nearly  on  the  same  par- 
allel of  declination. 

(317.)  The  Nautical  Almanac  furnishes  the  right  ascension 
of  the  moon's  bright  limb  for  the  lower  as  well  as  the  upper  cul- 
mination, L.  C.  being  put  to  denote  the  lower  culmination,  and 
U.  C.  the  upper  culmination.  The  right  ascension  of  the  moon's 
bright  limb  is  given  for  every  day,  with  a  view  to  the  more  ac- 
curate determination  of  its  variation,  when  required.  It  also 
furnishes  the  variation  in  right  ascension  of  the  moon's  limb  in 
one  hour  of  longitude  ;  that  is,  the  variation  during  the  interval 
of  her  transit  over  two  meridians,  equidistant  from  that  of  Green- 
wich, and  one  hour  distant  from  each  other.  These  numbers 
are  deduced  from  the  right  ascensions  of  the  bright  limb,  and 
therefore  include  the  effect  produced  by  the  change  of  the  semi- 
diameter. 

(318.)  These  numbers  enable  us  to  determine  the  difference 
of  longitude  of  any  two  places  where  corresponding  observations 
of  the  moon's  limb  have  been  made.  The  observations  furnish 
the  right  ascension  of  the  moon's  bright  limb  at  its  transit  over 
each  meridian,  which  we  will  represent  by  A  and  Ax ;  hence  we 
know  the  moon's  motion  in  right  ascension,  Ax— A,  during  the 
interval  of  the  two  transits.  But  the  Almanac  furnishes  the  va- 
riation of  the  moon's  right  ascension  corresponding  to  one  hour, 
which  we  will  represent  by  Y. 

We  shall  therefore  have  the  proportion 

Y  :  Ax  — A ::  1  hour  :  the  difference  of  longitude. 

Ex.  1.  The  right  ascension  of  the  moon's  first  limb,  Septem- 


314  PRACTICAL   ASTRONOMY. 

ber  6, 1840,  was  observed  at  Washington  to  be  19h.  21m.  29.90s., 
and  on  the  same  evening,  at  Hudson,  Ohio,  19h.  22m.  9.72s.  Re- 
quired the  difference  of  longitude  of  the  two  places. 

Here  Ax-  A  =  39.82s. 

That  value  of  Y  must  be  taken  which  corresponds  to  the  mid- 
dle of  the  interval  between  the  observations,  which  is  found  by 
interpolation  to  be  135.55s.  Hence  we  have 

135.55s. :  39.82s. ::  1  hour :  17m.  37.56s., 
which  is  the  required  difference  of  longitude. 

(319.)  Since  the  moon's  motion  in  right  ascension  is  not  uni- 
form, this  method  of  reduction  can  not  be  relied  upon  when  the 
distance  between  the  meridians  is  considerable.  The  following 
method  in  such  cases  is  to  be  preferred : 

Let  a)  represent  the  approximate  longitude  of  the  station  to 
be  compared  with  Greenwich,  as  "Washington,  for  example,  and 
6>H-  x  the  true  longitude  to  be  determined.  Let  A  and  A'  be  the 
observed  right  ascensions  of  the  moon's  limb  at  the  moments  of 
its  passing  the  meridians  of  Greenwich  and  Washington  respect- 
ively. These  will  evidently  be  the  sidereal  times  of  her  transit 
at  those  places.  Find,  by  interpolation  from  the  Nautical  Al- 
manac, the  moon's  right  ascension  for  the  assumed  longitude  w, 
and  call  it  A/x.  Now  A7,  the  sidereal  time  of  transit  of  the 
moon's  limb  at  Washington,  is  her  right  ascension  for  the  true 
longitude  a)  +  x,  and  consequently  Ax  — A/x  is  the  increase  of  the 
moon's  right  ascension  for  the  small  arc  of  longitude  x. 

Let  m/=A./  —  A  =  the  observed  increase  of  right  ascension  of 
the  moon's  limb  between  the  two  me- 
ridians. 

w  =  A/x  —  A  — the  increase  computed  for  the   assumed 
longitude  o>. 

Then  m'  —  m  —  hf— A7/  =  the  excess  of  the  observed  increase 
above  the  computed  increase. 

And  we  shall  have 

m-.G)'.:  m'—mix; 

that  is,  x  =  —  (m/  —  m). 

mv 

The  true  longitude  —  a>+x. 

Ex.  2.  The  increase  of  right  ascension  of  the  moon's  bright 
limb  between  her  transits  over  the  meridians  of  Greenwich  and 


LONGITUDE. 


315 


Hudson,  Ohio,  September  6,  1840,  was  found  to  be  12m.  17.95s. 
Required  the  difference  of  longitude. 

According  to  the  Nautical  Almanac,  the  right  ascension  of  the 
moon's  bright  limb  for  Greenwich  transit  was, 


Date. 

R.  A.  Moon's 
Limb. 

D'. 

D". 

D'". 

D"". 

h.    m.       s. 

m.       s. 

s. 

*. 

s. 

Sept.  5,  U.  C. 

18  14  45.36 

+  27  40.48 

L.  C. 

18  42  25.84 

-14.80 

+  27  25.68 

-5.77 

"     6,  U.  C. 

19     9  51.52 

^  =  —20.57 

«„=  +  !.  68 

60  =  +27     5.11 

d,  =  -4.09 

L.  C. 

19  36  56.63 

c0  =  -24.66 

«,  =  +  1.94 

+  26  40.45 

-2.15 

"     7,  U.  C.  i20     3  37.08 

-26.81 

+26  13.64 

L.  C. 

20  29  50.72 

and  the  successive  orders  of  differences  are  found  as  above. 


Hence  we  have,  by  Art.  223, 
+  1625.11s.      |      -22.615s. 


-409s. 


+L81s. 

We  will  assume  w  to  be  5h.  25m.  40s.     The  process  for  find- 
ing the  value  of  A"—  A,  by  Art.  223,  will  be  as  follows  : 


5h.  25m.  40s.  -    19540s.  -4.2909246 

12h.=   43200s. -4.6354837 

t=  .452315-9.6554409 

6-1625.11s.-3.2108828^ 

+735.061s. -2.8663237 

log.  Z -9.65544 
*-l  =  -.547685=9.73853» 
2  comp.-9.69897 


log.  *(*-l)=9.3940» 
J  =  -.047685=8.6784» 

6  comp. -9.2218 
d=-4.09s.=0.6117» 
-0.008s.  =7S)Q59n 
log.  *(*-l)=9.3940» 
-    1.452315-0.1621 


24  comp.  -8.6198 
e=+1.81s. -0.2577 
+0.042s. -8.6233 


+2.801s.-0.44734 
Hence 

A"  -  A = m  =  735.061s.  -+  2.801s.  -  .008s.  +  .042s.  =  737.896s. 
But  m'  =  737.95s.  =  the  observed  increase  of  right  ascension. 
Hence  m'  —  m=-\- 0.054s.  =  the  observed  excess. 
Therefore 

737.896s. :  19540s. ::  0.054s. :  x  =  1.43s. 

The  assumed  longitude  was 5h.  25m.  40s. 

The  correction  is -f  1.43s. 

The  longitude  from  this  observation  is .  .  5h.  25m.  41.43s. 


316  PRACTICAL   ASTRONOMY. 

(320.)  The  increase  of  right  ascension  of  the  moon's  bright 
limb  should,  if  possible,  be  derived  from  actual  observations  at 
Greenwich ;  or,  at  all  events,  the  errors  of  the  tables  should  be 
corrected  by  observations  at  some  standard  observatory. 

The  chief  disadvantage  of  this  method  consists  in  this  cir- 
cumstance, that  an  error  in  the  observed  increase  of  right  as- 
cension will  produce  an  error  between  20  and  30  times  as  great 
in  the  computed  longitude.  The  increase  of  right  ascension  of 
the  moon's  limb  in  one  hour  of  longitude  varies  from  112  seconds 
to  180  seconds.  In  the  former  case,  an  error  of  one  second  in 
the  observed  increase  of  right  ascension  would  cause  an  error 
of  32  seconds  in  the  deduced  longitude ;  and  in  the  latter  case, 
it  would  cause  an  error  of  20  seconds.  Hence,  to  obtain  a  sat- 
isfactory result  by  this  method,  requires  a  series  of  observations 
made  with  the  utmost  care,  and  continued  through  a  long  period 
of  time. 

(321.)  It  is  found  that  telescopes  of  different  optical  power  do 
not  exhibit  the  moon  of  the  same  diameter ;  and  the  determina- 
tion of  longitude  from  a  single  observed  moon-culmination  is  al- 
ways liable  to  error  from  this  source.  In  order  to  eliminate  this 
error,  we  should  so  arrange  the  series  of  observations  that  the 
error  shall  sometimes  be  in  excess,  and  a"t  other  times  in  defect ; 
and  this  is  accomplished  by  observing  successively  both  limbs  of 
the  moon ;  that  is,  by  observations  of  the  first  limb  before  full 
moon,  and  of  the  second  limb  after  full  moon.  The  observer 
should  also  take  care  that  the  apparent  diameter  of  the  moon  is 
not  magnified  by  imperfect  optical  adjustment  of  his  telescope  ; 
for  which  purpose  he  must  see  that  the  eye-piece  is  accurately 
adjusted  to  the  focus,  in  order  that  the  moon  and  the  spider  lines 
may  both  appear  sharp  and  distinct  at  the  same  time. 

(322.)  The  method  of  determining  longitude  by  lunar  dis- 
tances is  closely  allied  to  the  method  of  moon-culminating  stars  ; 
but  this  method,  being  little  used  in  fixed  observatories,  is  not 
treated  of  in  the  present  volume.  The  common  mode  of  re- 
ducing a  lunar  distance  yields  very  imperfect  results  ;  but  in  the 
American  Nautical  Almanac  for  1855,  Professor  Chauvenet  has 
given  a  method  of  making  the  reductions  with  entire  accuracy, 
and  has  furnished  tables  by  which  the  computations  are  made 
with  great  facility. 


LONGITUDE.  317 


SECTION   IV. 

LONGITUDE  DETERMINED  FROM  OCCULTATIONS  OF   STARS,  BY  FINDING 
THE    TIME    OF    TRUE    CONJUNCTION. 

(323.)  On  account  of  the  moon's  parallax,  it  often  happens 
that  a  star  which  is  occulted  by  the  moon  to  an  observer  at  one 
station  is  not  occulted  at  a  second  station,  or  the  occultation  be- 
gins at  a  different  instant  of  time.  We  can  not,  therefore,  use 
an  occultation  as  an  instantaneous  signal  for  comparing  directly 
the  local  times  at  the  two  stations ;  but  we  may  deduce  from 
the  observed  occultation  the  time  of  true  conjunction  of  the 
moon  and  star ;  that  is,  the  time  of  conjunction  as  seen  from  the 
centre  of  the  earth ;  and  this  is  a  phenomenon  which  happens 
at  the  same  absolute  instant  for  every  observer  on  the  earth's 
surface.  For  this  purpose,  we  must  determine  from  the  ob- 
served instant  of  immersion  or  emersion, 

1st.  The  apparent  difference  of  right  ascension  between  the 
moon  and  star. 

2d.  The  true  difference  of  right  ascension  between  the  moon 
and  star. 

3d.  The  time  of  true  conjunction. 

(324.)  In  the  annexed  figure,  let  P  represent 
the  pole  of  the  equator,  M  the  centre  of  the  moon, 
and  S  the  star  at  the  instant  of  immersion,  when 
its  apparent  distance  from  the  moon's  centre  is 
equal  to  the  moon's  semi-diameter.  Let  SD  be  a 
parallel  of  declination,  passing  through  S,  and  let 
AB  be  the  arc  of  the  equator,  intercepted  between  «g\ 
the  hour  circles  PS  and  PM  prolonged.  Then  MD 
is  the  apparent  difference  of  declinations  between 
the  star  and  the  moon's  centre,  which  we  will  rep-  ^ 

resent  by  6  ;  and  -  — —  is  their  apparent  difference  of  right  as- 
cos.  Ao 

censions,  which  we  will  represent  by  a.     Also,  we  will  represent 
SM,  the  moon's  semi-diameter,  by  A. 

Now  the  triangle  SMD  being  necessarily  very  small,  we  may 
regard  it  as  a  plane  triangle,  and  we  shall  have 


318  PRACTICAL    ASTRONOMY. 

Whence  a 

cos. 

putting  d  for  AS,  or,  more  properly,  J-(AS  +  BM). 

If  we  represent  the  moon's  parallax  in  right  ascension  by  TT, 
the  difference  between  the  true  right  ascensions  of  the  moon  and 
star  will  be  represented  by 


The  time  required  by  the  moon  to  describe  this  arc  may  be 
found  by  the  proportion 

m:  3600s.  ::a.±rr:t, 

where  m  is  the  hourly  motion  of  the  moon  in  right  ascension, 
corresponding  to  the  middle  of  the  interval  between  the  observed 
time  and  that  of  true  conjunction  of  the  moon  and  star. 

TJ  3600,         x 

Hence  t=.  --  (a  ±77). 

m 

Let  T  represent  the  observed  instant  of  immersion  or  emer- 
sion ;  then  T±t  will  be  the  instant  of  the  true  conjunction. 

(325.)  If  the  occupation  has  been  observed  under  a  second 
meridian,  we  may  in  the  same  way  determine  the  instant  of  true 
conjunction  at  the  second  place.  Now  the  absolute  instant  of 
this  phenomenon  is  the  same  for  both  places  ;  hence  the  differ- 
ence of  the  two  results  thus  obtained  is  the  difference  of  longi- 
tude of  the  two  stations.  If  the  two  stations  are  not  very  re- 
mote, the  effect  of  any  small  error  in  the  tables  of  the  moon  will 
be  partially  eliminated  from  the  result.  If  the  occultation  has 
not  been  observed  under  a  second  meridian,  we  must  calculate 
the  time  of  true  conjunction  for  Greenwich  according  to  the  ta- 
bles, and  compare  this  time  with  that  deduced  from  the  observa- 
tion. 

Example.  The  immersion  of  f]  Tauri  was  observed  at  the 
High  School  Observatory,  Philadelphia,  July  6,  1839,  at  16h. 
30m.  25.39s.  mean  time  ;  and  at  Hudson,  Ohio,  at  16h.  2rn. 
21.67s.  mean  time.  Required  the  difference  of  longitude  of  the 
two  places. 

"We  will  assume  the  longitude  of  Philadelphia  to  be  5h.  Om. 
42.5s.,  and  that  of  Hudson  to  be  5h.  25m.  41.3s.  ;  the  corre- 
sponding Greenwich  times  of  observation  will  be  21h.  31m. 
7.89s.,  and  21h.  28m.  2.97s. 

For  21h.  31m.  7.89s.  Greenwich  time,  the  moon's  equatorial 


LONGITUDE.  319 

parallax,  by  Adams'  Tables,  is  59'  41XX.7,  which,  reduced  to  the 
latitude  of  Philadelphia,  is  59X  36XX.8.     The  moon's  parallax  in 
right  ascension  for  this  case  was  computed  in  Ex.  1,  page  189, 
and  found  to  be  44X  17/x.l.     The  parallax  in  declination  was 
computed  in  Ex.  1,  page  194,  and  found  to  be  26'  10".l.     The 
moon's  true  semi-diameter  is  16X  16/x.O.     The  augmentation  was 
computed  in  Ex.  2,  page  201,  and  found  to  be  10X/.15.     Hence 
the  augmented  semi  -  diameter  is  16X  26//.2  =  A.     Also,  m^ 
2286//.2. 

The  moon's  true  decimation  .  .  =24°     5X  llx/.6  N. 
Parallax  in  declination  .....  =          26X  107/.l 

Moon's  apparent  declination  .  .  =23°  39X     lx/.5 
Star's  declination  ........  ^23°  36X  17x/.l 

6=  difference  ..........  =~~    ~~2X  44xx.4  =  164x/.4. 

Hence  A  +  rf=H50/x.6 

A-d=  821XX.8 

d=23°  37X  39XX.3 

a=  apparent  difference  of  R.  A.  =  v/1150'6  x  82L8  =  1Q61XX.9 

cos.  d 


?r  —  the  true  difference  of  right  ascension  .....  =3719/x.O 

3719.0x3600 

—  —  5856.2s.  =  lh.  37m.  36.2s., 


which,  added  to  16h.  30m.  25.39s.,  gives  18h.  8m.  1.59s.  for  the 
Philadelphia  time  of  true  conjunction. 

So  also  for  21h.  28m.  2.97s.  Greenwich  time,  the  moon's  equa- 
torial parallax  is  59X  41/x.7,  which,  reduced  to  the  latitude  of 
Hudson,  is  59X  36X  .5.     The  moon's  parallax  in  right  ascension 
for  this  case  was  computed  in  Ex.  3,  page  190,  and  found  to  be 
45X  56/x.5.     The  parallax  in  declination  was  computed  in  Ex.  3, 
page  195,  and  found  to  be  29X  17/X.9.     The  augmentation  of  the 
moon's  semi-diameter  was  computed  in  Ex.  4,  page  201,  and 
found  to  be  8XX.9.     Hence  the  augmented  semi-diameter  is  16X 
24XX.9=:A.     Also,  m  =  2286.1. 

The  moon's  true  declination  .......  =24°     4X  41/x.7 

Parallax  in  decimation  ..........  =          29X  17XX.9 

Moon's  apparent  decimation  .......  =23°  35X  23XX.8 

Star's  declination  .............  =23°  36X  17x/.l 

6=  difference  .  .  =  53XX.3 


320  PRACTICAL   ASTRONOMY. 


Hence  A  +  tf=1038".2 

A-d=  931/x.6 

d=23°  35X  50x/.4 

lA-tr  f-o    A      /1038.2x  931.6-   1fV7Q//0 

a  =  apparent  difference  of  it.  A.=  -  -  -  —  107  6"  2. 

cos.  a 

ir  =27  56".  5 

a+7r=the  true  difference  of  right  ascension    .  .  .  =  3829x/.7 

3829.7x3600     AmnQ       1T     ,n      Qn  Q 
t—  --  oooc^r  ~—  6030.8s.  =  lh.  40m.  30.8s., 


which,  added  to  16h.  2m.  21.67s.,  gives  17h.  42m.  52.47s.  for  the 
Hudson  time  of  true  conjunction. 

Subtracting  the  Hudson  time  of  true  conjunction  from  the 
Philadelphia  time,  we  obtain  25m.  9.1s.,  which  is,  therefore,  the 
difference  of  longitude  of  the  two  places,  as  determined  by  these 
observations. 


SECTION  V. 

LONGITUDE    DETERMINED    FROM    OCCULTATIONS    OF    STARS    BY   USING 
THE    MOON'S    MOTION    IN    ITS    APPARENT    ORBIT. 

(326.)  From  the  supposed  longitude  of  the  place  we  must  de- 
duce the  Greenwich  time  of  the  observation,  and  for  this  time 
find  the  true  place  of  the  moon,  and  compute  its  parallax  in  right 
ascension  and  declination,  from  which  we  derive  the  moon's  ap- 
parent place.  Subtracting  the  place  of  the  moon  from  that  of 
the  occulted  star,  we  obtain  the  apparent  distance  of  the  star 
.from  the  moon's  centre.  If  this  distance  is  equal  to  the  moon's 
semi-diameter,  augmented  for  its  apparent  altitude,  the  assumed 
longitude  is  correct ;  but  if  these  quantities  are  not  equal,  the 
assumed  longitude  is  erroneous,  and  the  correction  of  the  longi- 
tude may  be  obtained  according  to  the  principles  of  Section  III. 
of  Chapter  XL 

It  is  here  supposed  that  the  places  of  the  moon  given  in  the 
Nautical  Almanac  are  perfectly  correct.  In  order  that  the  longi- 
tude may  be  obtained  with  the  greatest  accuracy,  the  correc- 
tions of  the  tables  should  be  deduced  from  observations  at  some 
place  whose  longitude  is  well  known,  and  these  corrections  should 
be  applied  to  the  tabular  places  before  computing  the  distance 
between  the  moon  and  star. 


LONGITUDE. 


321 


Example.  The  immersion  of  a  Tauri  was  observed  at  Cam- 
bridge, January  23,  1850,  at  7h.  14m.  39.05s.  mean  time ;  the 
emersion  at  8h.  29m.  50.25s.  mean  time.  Required  the  longi- 
tude of  Cambridge  from  Greenwich. 

Assuming  the  longitude  of  Cambridge  from  Greenwich  to  be 
4h.  44m.  30s.,  the  corresponding  Greenwich  times  of  immersion 
and  emersion  will  be  llh.  59m.  9.05s.,  and  13h.  14m.  20.25s. 
For  these  times  we  find  the  right  ascension  and  declination  of 
the  moon  from  the  Nautical  Almanac,  and  apply  the  corrections 
found  on  page  340.  At  the  time  of  immersion,  the  moon's  hour 
angle  was  14°  46'  llx/.85  E. ;  its  horizontal  parallax,  reduced 
to  the  latitude  of  Cambridge,  was  59X  44".6  ;  and  its  semi-diam- 
eter, augmented  for  altitude,  16X  33/x.4.  At  the  time  of  emer- 
sion, the  hour  angle  was  3°  18'  12".  95  "W. ;  its  reduced  hori- 
zontal parallax,  59X  47X/.0 ;  and  its  augmented  semi-diameter, 
16X  34/x.6.  Hence  we  obtain  the  following  results : 


j              For  Immersion. 

For  Emersion. 

Moon's  true  place  
Correction  to  do  
Moon's  parallax  

R.  A.. 
h.  m.       s. 
4  25  34.12 
-0.52 
47.70 

,Dec;, 
16  29  58.5  N. 
-2.3 
26  43.5 

R.  A. 
h.  m.       s. 
4  28  40.02 
-0.52 
10.80 

Dec. 

0            '              // 

16  38     5.3  N. 
-2.3 
26  13.2 

Moon's  apparent  place  ... 
Star's  place  

4  26  21.30 
4  27  19.54 

16     3  12.7 
16  12     3.4 

4  28  28.70 
4  27  19.54 

16  11  49.8 
16  12     3.4 

Difference  

58.24 
839.5 

8  50.7 
530.2 

1     9.1& 
996.2 

13.6 
12.9 

Reduced  to  seconds  of  arc  . 

The  hourly  motion  in  right  ascension  is  l4647/.9,  arid  in  dec- 
lination 412/x.8. 

Hence,  in  the  triangle  HMMX  (see  fig.  next  page), 
14647/.9 ;  412x/.8 ::  1 :  tang.  HMMX=15°  44'  16".2, 

cos.  HMM' :  1 ::  1464x/.9 :  MM'  =  1521/x.97, 
which  is  the  hourly  motion  in  orbit. 
In  the  triangle  DSM, 

530x/.2 : 839x/.5 ::  1 :  tang.  DSM  =  57°  43'  33//.2, 

sin.  DSM :  839//.5 ::  1 :  SM  =  992x/.94. 
Hence  MSC  =  73°  27'  49".4, 

1 : 992/x.94 ::  cos.  MSC  :  SC  =  282//.6, 
SB=993//.4:282//.6::  1  :cos.  BSC^73°  28X  17//.8. 
Hence  BSM  =  28x/.4, 

sin.  SBM :  992/x.94 ::  sin.  BSM :  BM  =  Ox/.48. 
The  time  of  describing  BM= 1.14s. >  which  is  the  correction 

X 


322 


PRACTICAL   ASTRONOMY. 


to  the  assumed  longitude,  as  deduced  from  the  observed  im- 
mersion. 

m 


12". 9 : 996". 2 ::  1 :  tang.  DSMX^89°  15'  29/x.2, 

sin.  DSM' :  996".2 : :  1 :  SM'  =  996". 31. 
Hence  MXSC  =  73°  31'  13/x.O, 

994/x.6  :282/x.6 ::  1 :  cos.  ESC  =  73°  29X  31/x.7. 
Hence  ESM'  =  1' 41".3, 

sin.  SEM' :  996//.31 ::  sin.  ESM' :  EM'  =  1".72. 
The  time  of  describing  EMX  =  4.07s.,  which  is  the  correction 
to  the  assumed  longitude,  as  deduced  from  the  observed  emer- 
sion. 

Hence  the  longitude  of  Cambridge,  derived  from  the  observed 
Immersion,  is  .  .  4h.  44m.  30s.  —  1.14s.  ==  4h.  44m.  28.86s. 
Emersion,  is.  .  .  4h.  44m.  30s.  -4.07s.  =  4h.  44m.  25.93s. 

The  mean  of  the  two  results  is 4h.  44m.  27.40s. 

In  a  similar  manner,  we  may  determine  the  longitude  from  a 
solar  eclipse. 


SECTION   VI. 

BESSEL'S  METHOD  OF  COMPUTING  THE  LONGITUDE  OF  A  PLACE,  AND 
THE  ERROR  OF  THE  TABLES,  FROM  OBSERVATIONS  OF  A  SOLAR 

ECLIPSE. 

(327.)  We  obtain  directly  from  observation  either  the  sidereal 
time  or  the  apparent  solar  time  of  the  different  phases  of  an 


LONGITUDE.  323 

eclipse  ;  but  to  deduce  the  corresponding  mean  time  requires  at 
least  an  approximate  knowledge  of  the  longitude  of  the  place. 
We  may,  however,  generally  assume  that  the  longitude  is  known 
with  sufficient  precision  to  enable  us,  without  material  error,  to 
deduce  the  mean  time  from  the  known  sidereal  time,  or  apparent 
solar  time.  We  shall,  therefore,  suppose  that  both  the  sidereal 
time  and  the  mean  time  of  the  phases  of  an  eclipse  are  known, 
and  also  the  latitude  of  the  place  of  observation. 

(328.)  The  general  elements  of  the  eclipse  in  question,  A,  D, 
i,  /,  x,  y,  and  z,  must  be  computed  from  hour  to  hour  for  the 
mean  time  of  the  meridian  of  the  ephemeris,  and  these  hours 
must  be  so  selected  as  to  comprehend  the  entire  duration  of  the 
eclipse.  The  formulas  for  these  quantities  have  been  given  on 
page  277. 

The  values  of  A,  D,  and  i  change  but  slowly,  and  we  may  as- 
sume them  to  be  pretty  accurately  known  for  the  time  of  ob- 
servation ;  for  i  is  extremely  small,  while  A  and  D  depend  chiefly 
upon  the  place  of  the  sun,  which  the  tables  furnish  with  tolera- 
ble precision.  Indeed,  this  assumption  is  a  necessary  one,  for 
it  is  impossible  from  the  observations  of  an  eclipse  to  detect  any 
error  which  may  exist  in  these  values.  The  errors,  however, 
existing  in  the  assumed  values  of  x,  y,  and  /  may  be  determined 
with  great  accuracy  ;  and  we  shall  therefore  substitute  for  these 
quantities  the  expressions  rc  +  Aic,  y-\-ky,  /  +  A/,  where  A:E,  A#, 
and  A/  represent  the  errors  of  x,  y,  and  /,  of  which  we  are  in 
search.  Equation  (5),  page  267,  accordingly  becomes 


(329.)  Let  now  the  values  of  a,  d,  TT,  a',  d',  and  TT',  be  taken 
from  the  ephemeris  for  the  time  T  of  the  first  meridian.  Let 
T  +  T'  represent  the  required  time  of  the  first  meridian  at  which 
a  phase  of  an  eclipse  was  observed.  Let  x0  and  y0  denote  the 
values  of  x  and  y  for  the  time  T,  and  xf  and  y'  the  hourly  varia- 
tions of  x  and  y  ;  then  we  shall  have 

x  =  x0  +  x"£',  and  y=y0+y/T/. 

We  may,  in  the  same  manner,  consider  |,  ??,  and  £  as  also  com- 
posed of  two  parts.  Since,  however,  these  magnitudes  change 
but  slowly,  and  we  generally  have  an  approximate  knowledge 
of  the  difference  of  longitude,  and  consequently  the  time  of  the 
first  meridian  corresponding  to  the  time  of  observation,  we  may 


324  PRACTICAL    ASTRONOMY. 

assume  these  quantities  as  known  for  this  time.     The  preceding 
equation  therefore  becomes 


(330.)  If  the  variations  of  x  and  y  were  proportional  to  the 
time,  x/  and  y'  would  be  constant,  and  the  knowledge  of  the 
time  T  +  Tx  would  not  be  necessary  for  computing  them.  This, 
however,  is  not  the  case  ;  but  since  the  variations  of  x/  and  y' 
are  small  in  comparison  with  those  of  x  and  y,  the  above  equa- 
tion may  be  solved  by  successive  approximations,  which  rapidly 
converge  to  the  truth. 
Let  us  assume 

m  sin.  M  =  a;0—  |,  n  sin.  N=rc/, 

m  cos.  M=y0  —  77?  ncos.N  —  y', 

/-tf=L, 
the  preceding  equation  will  become 

(L  +  A/)3  =  (w  sin.  M  +  raT'  sin.  N+Az)2 
+  (m  cos.  M  +  raT'  cos.  N  +  Ay)2. 

Substitute  for  L  its  value  -  H  --  -,  and  expand  the  sec- 

sin.  ij 


ond  member  of  this  equation,  remembering  that  m2  sin.2  M  +  m 
cos.2  M=m2,  we  obtain 
/*  sin.  (M-N)     A/V  = 


oos.       - 
sm. 


sn.      +^      sn. 
cos.  M  +  ^TX  cos. 

cos.  (M-N)  +  A£  sin.  N+AT/  cos.  N}2, 
4-  [w  sin.  (M-N)  +  Aa;  cos.  N—  Ay  sin.  Nf2. 
Let  us  put         nX  —  &x  sin.  N  -f  Ay  cos.  N, 

n^=—^x  cos.  N+Ay  sin.  N, 
and  we  shall  have 

(«Bin.(M-N)  '=  {sT 

sm.  i> 


-r  -  sin.  (M- 
—  {m  sin.  (M-N)- 

=  -7^-  sin.2  (M-N)+  J^-  sin.  (M-N)A/ 
sm/  i/>  sin.  i/; 

-m2  sin.2  (M—  N)  +  2m»r  sin.  (M-N) 


LONGITUDE.  325 


sn.     , 
where  we  have  neglected  the  small  terms  A/2  and  n2k/z. 

Extracting  the  square  root,  and  neglecting  the  higher  powers 
of  A/  and  nX',  we  have 

•ttT'-f  mcos.  (M  —  N)  +  ^A  =  +  {Lcos.i/j+A/sec.i/j+rcA'tang.i/j}, 
or 

T'  =  --  cos.  (M-N)  +  L  COS"  ^-A  +  ^tang.  -0+—  sec.  -0, 
n  n  n 

m   sin.  (M  —  N±V>)  A/ 

or      T'  =  --  .  -  W-  —  —  —  '  —  A  =F  Ax  tang,  ib  =F  —  sec.  i/>. 
^  sin.  i/>  ^ 

(331.)  Since  now  the  time  of  immersion  is  always  earlier  than 
that  of  emersion,  T',  for  an  immersion,  must  have  a  less  positive 
value,  or  a  greater  negative  value,  than  for  emersion.  Hence, 
if  we  always  take  the  angle  i/>  either  in  the  first  or  fourth  quad- 
rant, the  upper  sign  belongs  to  an  immersion,  the  lower  to  an 
emersion.  If,  however,  for  an  immersion  we  take  i/;  in  the  first 
or  fourth  quadrant,  but  for  an  emersion  in  the  second  or  third 
quadrant,  we  shall  have  in  either  case, 
m  sin.  (M  — 


.       — 

T'=  ---  2L_  --  LJLZ—A—^tang.  ib  --  sec.  ib, 
n  sm.  ip  n 

or 

T=  --  cos.  (M-N)-L  COS-  ^-A-r  tang,  j-**  sec,  y,  .  (1) 
n  n  n 

In  the  case  of  annular  eclipses,  at  the  internal  contact  the 
emersion  precedes  the  immersion.  We  must,  therefore,  in  this 
case,  for  the  immersion,  take  i/>  in  the  second  or  third  quadrant, 
and  for  the  emersion  in  the  first  or  fourth. 

(332.)  Equation  (1)  may  be  solved  by  successive  approxima- 
tions. We  must,  for  this  purpose,  compute  the  values  of  2,  #,  3:, 
A,  D,  g*,  /,  and  /,  for  several  successive  hours,  so  that  the  values 
of  x0  and  2/0,  as  well  as  their  hourly  variations,  can  be  found 
for  any  time  by  interpolation.  We  then  assume  a  time,  T,  as 
accurate  as  the  provisional  knowledge  of  the  difference  of  longi- 
tude will  permit,  and  interpolate  for  this  time  the  quantities  rr0, 
#0,  x',  and  y',  and  thence  find,  by  formula  (1),  an  approximate 
value  of  Tx.  With  the  value  T  +  lv,  we  repeat,  if  necessary, 
the  preceding  computation.  Represent  by  T  the  value  assumed 
in  the  last  approximation,  and  the  correction  obtained  by  Tx; 
then  T  +  T7  —  t  —  w,  where  t  is  the  time  of  observation,  and  w  the 


326  PRACTICAL    ASTRONOMY. 

east  longitude  of  the  station  from  the  first  meridian,  by  which 
we  understand  that  meridian  whose  time  is  employed  in  the 
computation  of  x,  y,  z,  etc. 
"We  therefore  have 

w  =  *-T  +  -  cos.  (M-N)+-  cos.  V+A+A'  tang.  i/>  +  —  sec.  $ 

'il  'Yl  fl 

=  ,_T  +  "sin.(M-N+V,)  +  ;t+  A?  .  . 

n  sin.  i/>  n 

(333.)  Since  the  mean  solar  hour  has  been  employed  as  the 
unit  in  the  values  of  x'  and  ?/x,  the  preceding  formula  supposes 
the  same  unit  of  time  for  w.  If  we  wish  to  obtain  the  differ- 
ence of  longitude  in  seconds  of  time,  we  must  multiply  the  form- 
ula by  s,  the  number  of  seconds  belonging  to  an  hour  of  that 
species  of  time  in  which  the  observation  is  expressed  ;  t—  T  will 
then  be  expressed  in  seconds  of  the  same  kind  of  time  in  which 
t  is  given  ;  or  T  represents  the  same  kind  of  time  with  t. 

Equation  (2)  does  not  properly  furnish  the  difference  of  longi- 
tude of  the  place  of  observation  from  the  first  meridian,  but 
rather  the  relation  between  this  quantity  and  the  errors  of  the 
elements  employed.  If  the  same  eclipse  has  been  observed  at 
different  places,  we  may  obtain  for  each  place  as  many  such 
equations  as  there  are  instants  of  observation.  By  combining 
these  equations,  we  may  eliminate,  as  will  be  seen  hereafter,  the 
error  of  one  or  more  of  the  elements  of  computation,  and  thus 
render  the  result,  as  far  as  possible,  independent  of  the  error  of 
the  tables. 

(334.)  We  must  now  develop  the  quantities  A  and  Ax,  which 
are  determined  by  the  equations 


wA'  =  sin.  NAT/—  cos.  NAx. 

The  quantities  x  and  ?/,  as  will  be  seen  from  the  equations  on 
page  271,  depend  upon  a  —  A,  6—  D,  and  TT.     If  we  assume  these 
magnitudes  to  require  correction,  we  shall  have 
b,x  —  a  A(a—  A)  +  £  A((5  —  D)  +  c  ATT, 


where  a,  b,  c  are  the  differential  coefficients  of  x  in  respect  to 
a—  A,  6  —  D,  and  TT;  while  a',  b',  and  c'  are  the  same  differen- 
tial coefficients  of  y.  Since  A(a  —  A),  A((5  —  D),  and  ATT  are  very 
small  quantities,  in  the  expressions  for  the  differential  coeffi- 


•<  LONGITUDE.  327 

cients  we  may  neglect  the  terms  which  contain  sin.  (a—  A)  and 
sin.  (6  —  D)  as  factors,  and  assume  cos.  (a  —  A)  and  cos.  (6  —  D) 
as  equal  to  unity.  We  thus  obtain,  by  differentiating  the  val- 
ues of  x  and  y  on  page  271, 

cos.  6          .        .  x     cos.  6 

a  —  -  -  cos.  (a—  A)=-  --  , 

sin.  TT  sm.  TT 

,  _      sin.  6  sin.  (a—  A)_n 

O  -    --  ;  -  -  U, 

sm.  TT 

_     cos.  6  sin.  (a—  A)  cos.  TT_ 

sin.2  TT 
/_cos.  6  sin.  D  sin.  (a—  A)_n 

Cb    —•  -  -  -  -  —  -  —  U« 

sm.  TT 


^ 


sn.  TT         sin.  TT' 


tang.  TT 


(335.)  Since  now  /I  and  A7,  as  also  A(a—  A),  A(d—  D),  and  ATT 
are  expressed  in  parts  of  radius,  if  we  wish  to  obtain  the  errors 
of  the  elements  in  seconds,  these  differential  coefficients  must  be 
divided  by  206265.  Let  us,  then,  put 


h  = 


206265  .  n  sin.  TT' 
and  we  shall  have 

A  —li  sin.  N  cos.  tfA(a—  A)-f  A  cos.  NA((5—  D) 
—  h  cos.  7rA7r[a;  sin.  N+#  cos.  N], 

^=  —  A  cos.  N  cos.  ($A(a  —  A)  +  /i  sin.  NA(d—  D) 
+  A  cos.  TrATrfre  cos.  N—  y  sin.  N]. 

If  we  multiply  the  former  equation  by  cos.  i/>>  and  the  latter 
by  sin.  i/>,  then  add  the  two  equations  together,  and  divide  by  A, 
we  shall  obtain 

X  tang.  -0]  ^^-=sin.  (N-y)  cos-  <5A(a-A) 


+  cos.  (N- 
—  cos.  TTATT[^  sin.  (N—  ^)+y  cos.  (N— 

Hence  we  obtain  from  equation  (2),  page  326, 


328  PRACTICAL   ASTRONOMY. 

m      sin.  (M  —  N  +  VO  ,  >  sin.  (N—  ib) 
^--  -    —     !l        -  --'- 


,. 

.  cos.  <5A(a  —  A) 

n  sin.  ij)  cos.  i/> 

cos.  (N-0          p) 

COS.  ^) 

h  .  206265  sin. 


COS.  if) 

-h  cos. 


COS.  1/J  J 

(336.)  Every  observation  of  the  instant  of  an  eclipse  furnishes 
one  equation  of  the  preceding  form,  and  as  this  contains  five  un- 
known quantities,  five  such  equations  are  sufficient  for  their  de- 
termination. The  magnitudes  A/  and  ATT  can  not  generally  he 
determined,  unless  observations  are  made  at  places  widely  sep- 
arated from  each  other.  Nevertheless,  the  computation  of  the 
coefficients  will  always  show  what  influence  any  error  in  the 
values  of  TT  and  I  may  have  upon  the  result.  We  therefore  gen- 
erally seek  to  free  the  difference  of  longitude  only  from  the  er- 
rors of  a  and  6  ;  hut  the  value  of  Aa  can  not  he  determined  un- 
less we  know  the  longitude  of  one  of  the  stations  from  the  first 
meridian. 

(337.)  The  following  is  a  synopsis  of  the  preceding  results  : 

Compute  the  values  of  e,  A,  D,  and  g*,  also  the  co-ordinates 
#,  #,  and  z,  from  the  formulas,  page  277,  with  the  quantities 
i  and  /,  all  of  which  quantities  are  general  for  all  places  on  the 
earth. 

1  Compute  also  the  following  formulae  : 
%=p  cos.  <$>'  sin.  (JJL  —  A), 

rj=p  sin.  (j/  cos.  D  —  p  cos.  <}>'  sin.  D  cos.  (//—A), 

%—p  sin.  <f>'  sin.  D+p  cos.  $'  cos.  D  cos.  (p—  A), 

where  all  the  symbols  have  the  same  signification  as  on  page 

278,  except  p,  which  here  represents  the  observed  sidereal  time 

of  contact. 

i 

Let  T  represent  the  approximate  time  of  the  first  meridian, 
corresponding  to  the  phase  observed. 

Let  x0  represent  the  value  of  x  for  the  time  T  ; 
2/o  represent  the  value  of  y  for  the  time  T  ; 
x/  represent  the  hourly  variation  of  x  ; 
y'  represent  the  hourly  variation  of  y. 


LONGITUDE.  329 


m  sn.     .  —  XQ  — 
m  cos.  M  =  ?/0  — 
^  sin.  N=a;/, 
w  cos.  N=' 


sin.  V>  =  T-  sm-  (M—  N). 


For  the  first  contact,  i/>  must  be  taken  in  the  first  or  fourth 
quadrant  ;  for  the  last  contact,  in  the  second  or  third  quadrant. 


m  sn.       -     +  V>=     m  ^  (M     N\         cos. 


n  sn. 
Then 


cos.  cJAa- 


COS. 

where 


COS.  Ip 

-) 


oncogg  > 

206265  .  ^  sm.  TT 

s  =  3600  =  the  number  of  seconds  in  an  hour  ; 

t  =  observed  mean  time  of  contact. 

(338.)  Example.  On  the  28th  of  July,  1851,  occurred  an 
eclipse  of  the  sun,  which  was  observed  as  follows  : 

?w.          ,*_   ___       At  Konigsberg,  Prussia. 

Beginning  ...........  3h.  38m.  10.8s.  Konigsberg  m.  t. 

Beginning  of  total  darkness   .  4h.  38m.  57.6s.  " 

End  of  total  darkness    .  .  .  .  4h.  41m.  54.2s.  " 

End  .....  .;.  .  .V.  .  .  .  5h.  38m.  32.9s.  « 

At  Washington,  District  of  Columbia. 

Beginning  .......  7h.  21m.  31.2s.  A.M.  Washington  m.  t. 

End  ..........  8h.  50m.  38.0s.  « 

It  is  required  to  determine  the  error  of  the  tables  and  the 
longitude  of  Washington. 

The  general  co-ordinates  for  this  eclipse  have  already  been 
given  on  pages  280  to  285.  Our  first  object  is  to  deduce  the 
error  of  the  tables  from  the  observations  at  Konigsberg. 


330 


PRACTICAL   ASTRONOMY. 


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LONGITUDE. 


331 


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332 


PRACTICAL   ASTRONOMY. 

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LONGITUDE. 


333 


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I* 


334 


PRACTICAL   ASTRONOMY. 


Hence  we  obtain  the  following  equations  for  Konigsberg : 
w=lh.  22m.  48.20s.  + 1.6809  Aa-   .4118Ad, 
a)  =  lh.  22m.  46.97s.  + 1.7805  Aa  +   .2681  Ad, 
6>=lh.  22m.  44.24s.  + 1.5677  Aa- 1.1749 Ad, 
G>  =  lh.  22m.  38.64s.  + 1.6765  Aa-   .4341  Ad. 
If  we  assume  the  longitude  of  Konigsberg  equal  to  Ih.  22m. 
0.5s.,  we  shall  have 

0  =  47.70s.  + 1.6809 Aa-  .4118 Ad, 
0  =  46.47s.  + 1.7805  Aa-f  .2681  Ad, 
0  =  43.74s.  + 1.5677  Aa- 1.1749  Ad, 
0  =  38.14s.  + 1.6765  Aa-  .4341  Ad. 

These  equations  should  be  solved  by  the  method  of  least 
squares,  explained  in  Art.  239.  Multiplying  each  equation  by 
the  coefficient  of  Aa  in  that  equation,  and  taking  the  sum  of  all 
these  products,  we  obtain 

0=295.4297  +  11.2638  Aa-2.7844  Ad. 

Multiplying  each  equation  by  the  coefficient  of  Ad  in  that 
equation,  and  taking  the  sum  of  all  these  products,  we  obtain 

0  =  -  75.1291  -  2.7844  Aa  + 1.8102  Ad. 
Solving  these  equations  in  the  usual  manner,  we  find 
Ad=r  +1". 87  —  error  in  declination, 

cos.  dAa=— 25/x.77, 
or  the  error  in  right  ascension  =  —  27x/.26. 

Aa  in  the  above  equations  is  used,  for  the  sake  of  brevity,  in 
place  of  cos.  dAa.     The  error  in  right  ascension  is  multiplied  by 
cos.  d,  to  reduce  it  to  an  arc  of  a  great  circle. 
Computation  for  Washington. 


Beginning. 

End. 

t 

7h.  21m.  31.2s.  m.  t. 

8h.  50m.  38.0s.  m.  t. 

3h.  43m.  49.35s.  s.  t. 

5h.  13m.  10.79s.  s.  t. 

p 

55°  57'  20".2 

78°  IT  41x/.8 

<£/  =  38°42/24//.7 

log.  P=  9.9994302 

Assume  T 

0.5h. 

2.0h. 

fl£$ 

-71°  10'    8x/.2 

-48°  53/  20".  0 

D 

19°    4/59//.7 

19°    4'    9/x.O 

p  COS.  (J/ 

9.8917227 

9.8917227 

sin.  (fj,—  A) 

9.9761089^ 

9.8770463^ 

log-  | 

9.8678316^ 

9.7687690/2 

1 

-.737618 

-.587177 

LONGITUDE. 


335 


Beginning. 

End. 

p  sin.  (t/ 

9.7955437 

9.7955437 

cos.  D 

9.9754523 

9.9754892 

log-  (1) 

9.7709960 

9.7710329 

p  COS.  <// 

9.8917227 

9.8917227 

sin.  D 

9.5144703 

9.5141615 

cos.  (//—A) 

9.5089050 

9.8179099 

log.  (2) 

8.9150980 

9.2237941 

(1) 

+  .590196 

+  .590246 

(2) 

+  .082243 

+  .167415 

(l)-(2)  =  »j 

+  .507953 

+  .422831 

p  sin.  0' 

9.7955437 

9.7955437 

sin.  D 

9.5144703 

9.5141615 

log.  (3) 

9.3100140 

9.3097052 

p  COS.  (// 

9.8917227 

9.8917227 

cos.  D 

9.9754523 

9.9754892 

cos.  (p—  A) 

9.5089050 

9.8179099 

log.  (4) 

9.3760800 

9.6851218 

(3) 

+  .204180 

+  .204035 

3  +1  - 

+  .237728 

+  .484308 

+  .441908 

+  .688343 

log.  $ 

9.6453318 

9.8378049 

log.  « 

7.6632448 

7.6632477 

log^  f  « 

7.3085766 

7.5010526 

.534242 

.534181 

# 

.002035 

.003170 

l-i$=L 

.532207 

.531011 

X0 

-1.054056 

-0.199758 

1 

-0.737618 

-0.587177 

'   »o-| 

-0.316438 

+  0.387419 

2^o 

+  0.927049 

+  0.802116 

*? 

+  0.507953 

+  0.422831 

2/0-^7 

+  0.419096 

+0.379285 

log.  (a;0-|) 

9.5002887ft 

9.5881809 

log-  (y0-7?) 

9.6223135 

9.5789657 

tang.  M 

9.8779752^ 

0.0092152 

M 

322°  56'  43".5 

45°  36X  28/x.2 

sin.  M 

9.7800115ft 

9.8540438 

m 

9.7202772 

9.7341371 

336 


PRACTICAL   ASTRONOMY. 


Beginning. 

End. 

X' 

+  .569487 

+  .569544 

y' 

-.083015 

-.083555 

log.  x' 

9.7554838 

9.7555272 

log.  y' 

8.9191566ft 

8.9219724ft 

tang.  N 

0.8363272^ 

0.8335548ft 

N 

98°  17'  37/x.2 

98°  20X  45/x.8 

sin.  N 

9.9954341 

9.9953761 

n 

9.7600497 

9.7601511 

M-N 

224°  39'  6/7.3 

307°  15/  427/.4 

sin.  (M-N) 

9.8468292ft 

9.9008463ft 

m 

9.7202772 

9.7341371 

comp.  L 

0.2739194 

0.2748965 

sin.  i/> 

9.8410258ft 

9.9098799ft 

* 

316°    5X  41x/.3 

234°  21X    4/x.7 

M-N+V 

180°  447  47/x.6 

181°  36X  47x/.l 

sin.  (M  —  N+V>) 

8.1149272ft 

8.4494768ft 

m 

9.7202772 

9.7341371 

comp.  n 

0.2399503 

0.2398489 

cosec.  i/> 

0.1589742ft 

0.0901201ft 

s  =  3600 

3.5563025 

3.5563025 

log.  T' 

1.7904314 

2.0698854 

T/ 

-61.72s. 

-117.46s. 

Z_T-T' 

-5h.  7m.  27.08s. 

-5h.  7m.  24.54s. 

N-i/> 

142°  11'  55/x.9 

223°  59X  41".l 

5=3600 

3.5563025 

3.5563025 

comp.  206265 

4.6855749 

4.6855749 

comp.  n 

0.2399503 

0.2398489 

cosec.  TT 

1.7547617 

1.7545655 

A 

0.2365894 

0.2362918 

sin.  (N—  1/>) 

9.7874058 

9.8417300ft 

sec.  i/} 

0.1423731 

0.2344702ft 

0.1663683 

0.3124920 

coefficient  of  cos.  dAa 

+  1.4668 

+  2.0535 

h 

0.2365894 

0.2362918 

cos.  (N—  -0) 

9.8977056ft 

9.8569725ft 

sec.  if} 

0.1423731 

0.2344702ft 

0.2766681ft 

0.3277345 

coefficient  of  Ad 

-1.8909 

+  2.1268 

Hence   we   obtain    the    following   equations    for  Washing- 
ton: 

w=-5h.  7m.  27.08s. +  1.4668Aa-1.8909Ad, 
w=-5h.  7m.  24.54s. + 2.0535 Aa+ 2. 1268 Ad. 


LONGITUDE.  337 

Employing  the  values  of  cos.  6&a  and  Ad,  found  on  page  334, 
we  obtain 

a)=  -5h.  7m.  27.08s.  -37.79s.-  3.54s.  =  -5h.  8m.    8.41s. 

G>=-5h.  7m.  24.54s. - 52.91s.  +  3.98s.  =  -5h.  8m.  13.47s. 

The  mean  of  the  two  results  is 

w=-5h.  8m.  10.94s., 

which  is  the  longitude  of  Washington  from  Greenwich,  accord- 
ing to  the  observations  of  the  solar  eclipse  of  July  28,  1851. 


SECTION   VII. 

BESSEL'S  METHOD  OF  COMPUTING  THE  LONGITUDE  OF  A  PLACE  AND 
THE  ERROR  OF  THE  TABLES  FROM  AN  OBSERVED  OCCULTATION. 

(339.)  The  formulas  of  the  preceding  section  are  applicable  to 
an  occultation  of  a  fixed  star,  with  the  modifications  indicated 
in  Art.  295.  The  computation  of  e,  A,  D,  g1,  /,  and  I  is  dis- 
pensed with,  as  also  z  and  £.  We  must  compute  the  values  of 
x  and  y  from  the  formulae 
cos.  6  sin.  (a— A) 

X  =  ; , 

,-J  sm.  TT 

-  sin-  (<*-P)  c°s-2  i(a-A)  +  sin.  (6+D)  sin.2  j(a-A) 

sin.  TT 

Also  the  values  of  £  and  ??  from  the  formulae 
£— p  cos.  $'  sin.  (ft— A), 

?j=p  sin.  0'  cos.  D—p  cos.  <//  sin.  D  cos.  (p— A), 
where  /^  represents  the  observed  sidereal  time  of  immersion  or 
emersion. 

Let  T  represent  the  approximate  time  of  the  first  meridian, 
corresponding  to  the  phase  observed. 

Let  x0  represent  the  value  of  x  for  the  time  T  ; 
yQ  represent  the  value  of  y  for  the  time  T  ; 
x'  represent  the  hourly  variation  of  x  ; 
y'  represent  the  hourly  variation  of  y. 
m  sin.  M=rc0— ft 
m  cos.  M=#0— 77, 
n  sin.  N=af, 
n  cos.  N— y', 
Y 


338 


PRACTICAL   ASTRONOMY. 


sn. 


777 

=  —  sin.  (M  —  N), 

K 


log.  k  =9.4353665. 

For  immersion,  i/>  must  be  taken  in  the  first  or  fourth  quad 
rant  ;  for  emersion,  in  the  second  or  third  quadrant. 


sin.  if) 


=_     cos. 

n 


n 


Then 


where 


COS.  l/> 


cos. 


COS. 


206265.  n  sin.  TT 
s  =  3600  =  the  number  of  seconds  in  an  hour ; 
t  =  the  observed  mean  time  of  immersion  or  emersion. 
(340.)  Example.  On  the  23d  of  January,  1850,  the  occulta- 
tion  of  a  Tauri  was  observed  as  follows  : 

At  Greenwich,  England. 

Immersion 13h.  32m.  38.66s.  Greenwich  m.  t. 

Emersion 14h.    1m.  24.52s.  " 

At  Cambridge,  Massachusetts. 

Immersion 7h.  14m.  39.05s.  Cambridge  m.  t. 

Emersion    .  ....  r.r,  .  .     8h.  29m.  50.25s.  " 

It  is  required  to  determine  the  longitude  of  Cambridge  from 
Greenwich. 

We  will  first  determine  the  error  of  the  tables  according  to  the 
Greenwich  observations.  The  co-ordinates  x  and  y  have  already 
been  given  on  page  294. 

Computation  for  Greenwich, 

^        ^1  O    1  "*}/  O  \  '/  ft 
G)    =Olu    I/     60:     .U. 


Immersion. 

Emersion. 

h.      m.          s. 

h.       m.         s. 

t 

13  32  38.66  m.  t. 

14     1  24.52  m.  t. 

Vt 

9  44  43.78  s.  t. 

10  13  34.37  s.  t. 

A 

4  27  19.54 

4  27  19.54 

P-A 

5  17  24.24 

5  46  14.83 

In  arc 

79°  21'  3x/.6 

86°  33'  42/x.45 

XQ 

.503232 

.787997 

y0 

.467948 

.519616 

X' 

.593999 

.593978 

y' 

.107785 

.107762 

LONGITUDE. 


339 


Immersion. 

Emersion. 

p 

9.9991134 

9.9991134 

COS.  0' 

9.7961416 

9.7961416 

sin.  (/£—  A) 

9.9924552 

9.9992176 

log.  | 

9.7877102 

9.7944726 

.613353 

.622978 

P 

9.9991134 

sin.  <// 

9.8922744 

cos.  D 

9.9824020 

log-  (1) 

9.8737898 

p  cos.  07 

9.7952550 

9.7952550 

sin.  D 

9.4456150 

9.4456150 

cos.  (//  —  A) 

9.2666830 

8.7779489 

log.  (S) 

8.5075530 

8.0188189 

(1) 

.747807 

.747807 

(2) 

.032178 

.010443 

.715629 

.737364 

X0      g 

-.110121 

+  .165019 

2/0  —  7? 

-.247681 

-  .217748 

log.  (XQ  —  %) 

9.0418701^ 

9.2175340 

log.  (Vo-??) 

9.39389277Z 

9.3379542^ 

tang.  M 

9.6479774 

9.8795798^ 

M 

203°  58X  13x/.2 

142°  50X  36x/.8 

sin.  M 

9.6088078?z 

9.7810323 

m 

9.4330623 

9.4365017 

log.*' 

9.7737857 

9.7737703 

log.  y' 

9.0325583 

9.0324656 

tang.  N 

0.7412274 

0.7413047 

N 

79°  42X  54".8 

79°  43X  l/x.2 

sin.  N 

9.9929653 

9.9929678 

% 

9.7808204 

9.7808025 

M-N 

124°  15'  18x/.4 

63°  T  35x/.6 

sin.  (M-N) 

9.9172630 

9.9503684 

m 

9.4330623 

9.4365017 

comp.  & 

0.5646335 

0.5646335 

sin.  if) 

9.9149588 

9.9515036 

ib 

55°  18"    7x/.2 

116°  34X  33x/.5 

M-N+V 

179°  33'  25x/.6 

179°  42X    9/x.l 

sin.  (M  —  N+i/>) 

7.8881678 

7.7153218 

m 

9.4330623 

9.4365017 

comp.  % 

0.2191796 

0.2191975 

cosec.  ip 

0.0850412 

0.0484964 

s_ggQQ 

3.5563025 

3.5563025 

10g.-T' 

1.1817534 

0.9758199 

-15.20s. 

-9.46s. 

340 


PRACTICAL   ASTRONOMY. 


Immersion. 


Emersion. 


N  —  V> 

24°  24'  47/x.6 

-36°  51X  32/x.3 

5=3600 

3.5563025 

3.5563025 

comp.  206265 

4.6855749 

4.6855749 

comp.  n 

0.2191796 

0.2191975 

cosec.  TT 

1.7589910 

1.7588792 

h 

0.2200480 

0.2199541 

sin.  (N  —  ip) 

9.6162807 

9.7780408ft 

sec.  i/> 

0.2446963 

:               0.3493195ft 

0.0810250 

0.3473144 

coefficient  of  cos.  dAo. 

+  1.2051 

+  2.2249 

A 

0.2200480 

0.2199541 

cos.  (N  —  1/>) 

9.9593220 

9.9031521 

sec.  i/; 

0.2446963 

0.3493195ft 

0.4240663 

0.4724257ft 

coefficient  of  Ad 

+  2.6550 

-2.9677 

Hence  we  have  the  two  equations, 

0  =  15.20  +  1.2051  Aa  +  2.6550Ad, 
0=  9.46  +  2.2249  Aa-  2.9677  Ad. 

From  which  we  obtain 

cos.  dAa=:-7//.405, 


Computation  for  Cambridge. 


Immersion. 

Emersion. 

A.     m.          s. 

!  h.     m.          s. 

t 

7  14  39.05  m.  t. 

8  29  50.25  m.  t. 

^ 

3  26  28.81  s.  t. 

4  41  52.36  s.  t. 

A 

4  27  19.54 

4  27  19.54 

p-A. 

-1     0  50.73 

14  32.82 

In  arc 

-15°  I8rt40".95 

3°  38X  12/x.3 

Assume  T 

12h. 

13.25h. 

X0 

-.413936 

+  .328553 

Vo 

+  .301478 

+  .436250 

x' 

+  .593945 

+  .593997 

y' 

+  .107833 

+  .107795 

p  COS.  0X 

9.8691208 

9.8691208 

sin.  (p—  A) 

9.4189320^ 

8.8022991 

log.  1 

9.2880528^ 

8.6714199 

t 

-.194112 

+  .046927 

LONGITUDE. 


341 


Immersion. 

Emersion. 

p  sin.  0X 

9.8264412 

cos.  D 

9.9824020 

log-  (1) 

9.8088432 

p  COS.  0X 

9.8691208 

9.8691208 

sin.  D 

9.4456150 

9.4456150 

cos.  (jit—  A) 

9.9845113 

9.9991246 

log.  (2) 

9.2992471 

9.3138604 

(1) 

+  .643937 

+  .643937 

(2) 

+  .199181 

+  .205997 

+  .444756 

+  .437940 

£0  —  £ 

-.219824 

+  .281626 

y0  —  rj 

-.143278 

-.001690 

log.  (iC0  —  |) 

9.3420751^ 

9.4496727 

log-  (Vo-??) 

9.1561795^ 

7.2278867'ft 

tang.  M 

0.1858956 

2.2217860ft 

M 

236°  54X  15/x.5 

90°  20X  37/x.8 

sin.  M 

9.9231195ft 

9.9999922 

m 

9.4189556 

9.4496805 

log.  xx 

9.7737463 

9.7737842 

log.  #x 

9.0327517 

9.0325986 

tang.  N 

0.7409946 

0.7411856 

N 

79°  42X  35XX.3 

79°  42X  51XX.3 

sin.  N 

9.9929578 

9.9929640 

n 

9.7807885 

9.7808202 

M-N 

157°  llx  40/x.2 

10°  37X  46x/.5 

sin.  (M-N) 

9.5883883 

9.2658995 

m 

9.4189556 

9.4496805 

comp.  k 

0.5646335 

0.5646335 

sin.  -0 

9.5719774 

9.2802135 

-0 

21°  54X  54xx.O 

169°    0X35XX.6 

M-N  +  0 

179°    6X34/X.2 

179°  38X  22xx.l 

sin.  (M  —  N+-0) 

8.1914938 

7.7988132 

m 

9.4189556 

9.4496805 

comp.  n 

0.2192115 

0.2191798 

cosec.  -0 

0.4280226 

0.7197865 

5  =  3600 

3.5563025 

3.5563025 

log.  Tx 

1.8139860 

1.7437625 

Tx 

-65.16s. 

-55.43s. 

w=£-T-Tx 

-4h.  44m.  15.79s. 

-4h.  44m.  14.32s. 

N—  0 

57°  47X  41XX.3 

-89°  17X44XX.3 

5  =  3600 

3.5563025 

3.5563025 

comp.  206265 

4.6855749 

4.6855749 

comp.  n 

0.2192115 

0.2191798 

cosec.  IT 

1.7593527 

1.7590595 

342 


PRACTICAL   ASTRONOMY. 


Immersion. 

Emersion. 

h 

0.2204416 

0.2201167 

sin.  (N—  1/>) 

9.9274447 

9.9999672^ 

sec.  ij) 

0.0325744 

0.0080389» 

0.1804607 

0.2281228 

coefficient  of  cos.  dAa 

4-1.5152 

+  1.6909 

A 

0.2204416 

0.2201167 

cos.  (N—  ijj) 

9.7266889 

8.0896618 

sec.  i/) 

0.0325744 

0.0080389^ 

9.9797049 

8.3178174^ 

coefficient  of  Ad 

+  .9543 

-.0208 

Hence  we  obtain  the  following  equations  for  Cambridge : 
G)=—  4h.  44m.  15.79s.  +  1.5152Aa  +  . 9543 Ad, 
a)=_4h.  44m.  14.32s.  + 1.6909 Aa-. 0208 Ad. 

Employing  the  values  of  Aa  and  Ad,  found  on  page  340,  we 
obtain 

G,=  -41i.  44m.  15.79s.  -11.22s.-  2.26s.  =  -4h.  44m.  29.27s. 
w=-4h.  44m.  14.32s.-  12.52s.  +  0.05s.  =  -4h.  44m.  26.79s. 

The  mean  of  the  two  results  is 

o>=-4h.  44m.  28.03s., 

which  is  the  longitude  of  Cambridge  from  Greenwich,  accord- 
ing to  the  observations  of  the  occultation  of  a  Tauri,  January 
23,  1850. 

(341.)  Ex.  2.  On  the  15th  of  April,  1850,  the  occultation  of 
a  Tauri  was  observed  as  follows  : 

At  Konigsberg,  Prussia. 

Immersion lOh.  57m.  43.7s.  Konigsberg  m.  t. 

Emersion llh.  47m.  47.6s.  " 

At  Cambridge,  Massachusetts. 

Immersion 2h.  1m.  52.45s.  Cambridge  m.  t. 

Emersion,  .i^...  .  .  .  3h.  1m.  38.35s.  " 

It  is  required  to  determine  the  longitude  of  Cambridge. 
According  to  the  Nautical  Almanac,  the  following  are  the 
places  of  the  moon  necessary  for  this  computation : 


LONGITUDE. 


343 


Greenwich  m.  t. 

i      t. 

7T. 

h. 

h.  m.    s. 

0               " 

6 

4  23  45.41 

+  16  40  0.1 

58  55.22 

7 

4  26  10.13 

16  46  30.5 

58  55.87 

8 

4  28  35.06 

16  52  54.7 

58  56.50 

9 

4-31  0.20 

16  59  12.7 

58  57.12 

10 

4  33  25.53 

17  5  24.4 

58  57.72 

11 

4  35  51.06 

17  11  29.7 

58  58.32 

According  to  Mr.  Adams'  tables,  each  of  the  preceding  paral- 
laxes should  be  increased  by  5/x.l. 
The  position  of  a  Tauri  was 

A=r4h.  27m.  18.26s.;  D  =  +  16°  12'  l/x.7. 

The  sidereal  time  at  Greenwich  mean  noon,  April  15,  was 

Ih.  33m.  8.96s. 


RECAPITULATION.  345 

The  following  recapitulation  of  the  formulae  most  frequently 
used  in  an  observatory  is  added  for  convenience  of  reference. 

To  compute  the  corrections  to  be  applied  to  the  observed 
transit  of  a  star,  in  order  to  obtain  the  correct  apparent  right 
ascension.  See  page  73. 

T>     A         m   ,    ^  ,          sm-  z       7     cos-  z  ,        c 

R.  A..  =  T  +  dt+a. h&. h 


cos.  6         cos.  6     cos.  d 

By  a  close  circumpolar  star  (see  page  69), 
«  =  JA  sec.  0  cot.  d. 

By  two  stars  differing  considerably  in  declination  (see  page 
70), 

A'  cos.  6  cos.  d' 

a  = . 

cos.  <f)  sin.  (6'  —  6) 

R.  A.=:the  apparent  right  ascension  required; 

T^the  observed  time  of  transit,  as  shown  by  the  clock ; 
eft = the  correction  for  the  error  of  the  clock,  plus  when  the 

clock  is  too  slow ; 
2= the  zenith  distance  of  the  star ; 
(5= the  declination  of  star  observed ; 
(5'= the  declination  of  the  second  star; 
#=the  deviation  of  the  telescope  in  azimuth,  plus  when 

the  eastern  pivot  deviates  to  the  north  of  east. 
b  —  the  inclination  of  the  axis  of  the  telescope  (see  page  63), 

plus  when  the  west  end  of  the  axis  is  too  high ; 
c  —  the  error  in  collimation  (see  page  65),  plus  when  the 

mean  of  the  wires  falls  on  the  east  side  of  the  optical 

axis; 
A  =  the  interval  between  two  successive  transits,  minus  12 

hours ; 
A'  =  the  difference  of  the  observed  times,  minus  the  difference 

of  right  ascensions ; 
0=the  latitude  of  the  place  ; 

z  =  <j>—6,  if  the  observations  be  made  to  the  south ; 
z  =  6— 0,  if  to  the  north,  above  the  pole: 
z=lSQ°  —  (0  +  <5),  if  to  the  north,  below  the  pole. 


346  PRACTICAL    ASTRONOMY. 

To  find  the  altitude,  azimuth,  and  parallactic  angle  of  a 
star,  its  declination  and  hour  angle  being  given,  as  well  as 
the  latitude  of  the  place.  See  pages  108  and  110. 

->,v.-.  •  '• 

tang,  y  =  cos.  P  cot.  6 ; 

sin.  6  sin.  (y  +  0)  a 
cos.  y 

cos.  z  cos.  y 
sin.  (?/  +  <£)—  —  -J  ; 

sin.  (5 


.      cot.  P  cos.  (y+d>) 

Cot.  A  =  i~_LIZ  ; 

sm. 


sin.     = 


sin. 


A=the  azimuth  of  the  star,  counted  from  the  north; 

z  =  th&  zenith  distance  of  the  star  ; 

P  =the  hour  angle  of  the  star  ; 

6  =  the  decimation  of  the  star  ; 

0  =  the  latitude  of  the  place  ; 

p  =  the  parallactic  angle. 

When  only  p  is  required, 

tang.  x=cos.  P  cot.  0; 

sin.  x  tansr.  P 
tang,  p  =  --  -  —  . 
cos.  (x  +  6) 


To  compute  the  distance  between  two  stars  whose  right  as- 
censions and  declinations  are  known.     See  page  111. 

cot.  B  — cos.  (a  —  ax)  cot.  d', 

cos.  (6/  —  B)  sin.  d 

cos.  x  = 2 — . — '       — . 

sin.  B 

a=the  right  ascension  of  one  star ; 
d=its  declination ; 

o!  =  the  right  ascension  of  the  second  star ; 
6/  =  its  declination ; 
#  =  the  angular  distance  required. 


RECAPITULATION.  347 

To  compute  the  hour  angle  at  the  pole :  the  latitude  of  the 
place,  the  declination,  and  zenith  distance  of  the  sun  or  star 
being1  given.  See  page  133. 


P  =  the  hour  angle  at  the  pole ; 
$=the  latitude  of  the  place ; 
d=the  declination  of  the  star ; 
j^^the  true  zenith  distance  of  the  star. 


To  compute  the  correction  for  the  reduction  to  the  meridian. 
See  page  142. 


__2  sin.2  -|P  cos.  0  cos.  6     ^sin.2  JP  cos. 

cc  ^^  —  -  — 


cos.  0  cos.  d\2  2  cot.  z 
in.  2:  /    sin.  lx/ ' 


P— the  hour  angle  at  the  pole,  as  shown  by  a  well-regulated 

clock ; 

0  =  the  latitude  of  the  place  ; 
<5= the  declination  of  the  star  ; 
z=the  meridional  zenith  distance  of  the  star; 
x=tln.Q  required  correction  in  seconds. 


To  compute  the  latitude  of  a  place,  from  observations  of  the 
pole  star  at  any  time  of  the  day.     See  page  152. 


(j)=H.-d  cos.  P  +  J  sin.  l/x(G?sin.  P)2  tang.  H 
-  J  sin.2  l/x  (d  cos.  P)  (d  sin.  P)2. 

H=:the  observed  altitude  of  the  star,  corrected  for  refraction  ; 
G?=the  apparent  polar  distance  of  the  star,  expressed  in  seconds 

of  arc  ; 

P  =  the  hour  angle  of  the  star  from  the  meridian  ; 
</>—  the  latitude  required. 


348  PRACTICAL   ASTRONOMY. 

To  find  the  altitude  and  hour  angle  of  a  star  when  it  is 
upon  the  prime  vertical,  together  with  the  latitude  of  the  place. 
See  pages  113  and  157. 

cos.  P  =  cot.  <j>  tang.  6  ; 

sin.  6 

sin.  A=-  -  ; 
sin.  </> 

cos.  A^sin.  P  cos.  6  ; 
tang,  d 


sin.  6 
sin.  (b=- 


sin.  A ' 
cos.  </>  =  cot.  P  cot.  A. 

P=the  hour  angle  of  the  star  at  the  pole ; 
A = the  altitude  of  the  star  ; 
(5  =  the  declination  of  the  star; 
0:=the  latitude  of  the  place. 


To  find  the  longitude  and  latitude  of  a  star  when  its  right 
ascension  and  declination  are  known,  and  vice  versa.  See 
pages.  174  and  176. 

Make  tang.  a  =  sin.  R.  A.  cot.  Dec. 

tang.  L  — sin.  (a  +  w)  tang.  R.  A.  cosec.  a, 
tang.  /  — cot.  (a-\-(*))  sin.  L, 
sin.  l=cos.  («  +  G))  sin.  Dec.  sec.  a, 
sin.  L  =  tang,  (a  -f  w)  tang.  /. 

Make  tang.  a  =  sin.  L  cot.  /. 

tang.  R.  A.=sin.  (a  —  w)  tang.  L  cosec.  a, 
tang.  Dec.  =  cot.  (a— w)  sin.  R.  A. 
sin.  Dec.  =  cos.  (a— w)  sin.  /  sec.  a, 
sin.  R.  A.  =  tang.  (a  —  u>)  tang.  Dec. 

L  =  the  longitude  of  the  star ; 
/  —  the  latitude  of  the  star  ; 
R.  A.  — the  right  ascension  of  the  star ; 
Dec.  =  the  declination  of  the  star; 
w  —  the  obliquity  of  the  ecliptic. 


RECAPITULATION.  349 

To  compute  the  longitude,  right  ascension,  and  declination 
of  the  sun  ;  any  one  of  these  quantities,  together  ivith  the 
obliquity  of  the  ecliptic,  being  given.  See  page  178. 

tang.  R.  A.  =  tang.  Long.  cos.  w  ; 
sin.  R.  A.  —  tang.  Dec.  cot.  w  ; 
tang.  Dec.  =  sin.  R.  A.  tang.  G>; 
sin.  Dec.  =  sin.  Long.  sin.  w  ; 
tang.  R.  A. 
--- 


sin.  Dec. 
sm. 


sin.  w 
cos.  Long.  =  cos.  R.  A.  cos.  Dec.  ; 

T,          cos.  Long. 
cos.  R.  A.  =  -------  TS-^-; 

cos.  Dec. 


Long.  =  the  sun's  longitude; 
R.  A.  =  the  sun's  right  ascension  ; 
Dec.  =  the  sun's  declination; 

6>=  the  obliquity  of  the  ecliptic. 


To  compute  the  correction  in  time,  to  be  applied  to  the 
mean  of  the  times  of  observed  equal  altitudes  of  the  sun,  in 
order  to  obtain  the  time  of  its  meridional  passage.  See  page 

128. 

cfP=T-(tang.  0  cosec.  P— tang.  6  cot.  P). 
15 

dP=the  increase  of  the  hour  angle  in  time ; 

dd— increase  of  declination  from  the  meridian  to  the  afternoon 

observation ; 

P=hour  angle  from  the  meridian,  supposing  no  change  in  dec- 
lination ; 

6= decimation  of  the  sun  on  the  meridian ; 
0=the  latitude  of  the  place. 


350  PRACTICAL    ASTRONOMY. 

For  the  Angle  of  the  Vertical. 
tang.  0'  =  tang.  0  x  0.9933254. 

For  the  Radius  of  the  Earth. 


COS.  <//  COS.   ((//  —  0) 

For  the  horizontal  Parallax  of  any  Place. 

For  the  Moon's  Parallax  in  Altitude. 

1.  sin.  q  =  sin.  p  sin.  (z  +  #). 

sin.  »  sin.  z 

2.  tang.  0=- if 

l  —  sm.  p  cos.  z 

sin.  7?  sin.  z  ,  sin.2  »  sin.  2z  ,  sin.3  jo  sin.  3z 


,,  , 
Make  a  = 


in.l-        -    sin.  8-    ~'W  -' 

e  Moon's  Parallax  in  Right  Ascension. 

sin.  j»  cos.  0X 

---  ~  --  £.. 

cos.  d 

1.     sin.  IL  =  a  sin. 

a  sin.  h 


2.  tang.  n= 


. 
—  a  cos.  h 

a  sin.  h     a2  sin.  2/i     a3  sin. 


Moon's  Parallax  in  Declination. 


1.  tang.  d,  =  /i_8in.fsin.»'\  sin.^ 
V  sin.  (5      /  sin.  h 


Make  cot.  6=oos.  (A+jn)  cot.  ^ 
cos.  Jn 


_sn.  ^?  sn.  ^ 
sin.  ^ 


RECAPITULATION.  351 

2.  sin.  TT=:£  sin.  (b  —  d  +  rr). 

c  sin.  (b  —  6) 

3.  tang.  TT=  v-      ->•— r,--'— T. 

1  —  c  cos.  (£  — d) 

.          c  sin.  (£— d)     c2  sin.  2(b  —  6)     c3  sin.  3(b  —  6) 
'  7r=      sin.  1"  smTS77"          "sinr^        ^' 6  °* 


hourly  Variation  of  Parallax  in  Right  Ascension. 

,     _J9  COS.  <//  COS. 

cos.  d 


For  the  hourly  Variation  of  Parallax  in  Declination. 
d-rr=p  cos.  <//  sin.  d  sin.  hdh. 

For  the  Augmentation  of  the  Moon's  Semi-diameter. 

1.  z  =  A .  s2  cos.  */  +  iAV  +  iA2s3  cos.2  ^  +  ,  etc., 
where  A =0.00001779. 

2.  x  =  s  sin.  TT  cot.  (6  —  (5)  —  %s  sin.27r. 

Notation. 

</>=the  geographical  latitude  of  the  place  ; 
(T/  —  the  geocentric  latitude  of  the  place  ; 
r  —  the  radius  of  the  earth,  corresponding  to  the  latitude  0 ; 
z  =  the  true  zenith  distance  of  the  moon  ; 
P  — the  moon's  horizontal  parallax  at  the  equator; 
p  =  the  horizontal  parallax  at  the  place  of  observation ; 
q  =  the  moon's  parallax  in  altitude ; 
d  =  the  true  declination  of  the  moon; 
6'  =  the  apparent  declination  of  the  moon ; 
h  =  the  true  hour  angle  at  the  pole  =  the  sidereal  time,  minus 

the  moon's  true  right  ascension ; 
h'  =  the  apparent  hour  angle  ; 
II  =  the  moon's  parallax  in  right  ascension  ; 
TT^the  parallax  in  declination  ; 
^n=the  variation  of  parallax  in  right  ascension; 
c?rr=:the  variation  of  parallax  in  declination ; 
5  =  the  true  semi-diameter  of  the  moon. 


352 


PRACTICAL   ASTRONOMY. 


TRIGONOMETRICAL    FORMULA. 


Values  of  sin.  A. 

Values  of  cos.  A. 

1 

sin.  A 

0 

COS.  A 

tang.  A 
sin  A  cot  A 

cot.  A' 

3. 

Vl  —  cos.2  A. 
1 

Vl-sin.2A. 
1 

. 

c 

Vl+cot.2A 
tang.  A 

Vl  +  tang.2  A 
cot.  A 

O. 

6. 

Vf+tang.2A* 
2  sin..  JA  cos.  JA. 

Vl  +  cot.2A 
cos.2  JA—  sin.2  ^A. 

7. 

/I  -cos.  2  A 
V     -%- 

2  tang.  JA 

1-2  sin.2  i  A. 
2cos.2iA-l. 

. 

l+tang.^A' 

2 

/1  +  cos.  2A 

. 
1  n 

tang.  iA+cot.  ^A' 
1 

l-tang.2£A 

lu. 

cosecant  A' 

cot.  ^  A—  tang.  JA 

11. 
19 

2  sin."  (45°  +  JA)  —  1. 
1     2  sin  2  (45°     AA^ 

cot.  JA+tang.  £A 
1 

JL/C. 

13. 

1+  tang.  A  tang.  ^A" 

1 

secant  A* 

Values  of  tang.  A. 

sin.  A 
cos.  A' 

1 

cot.  A 

V  cos.2  A 

sin.  A 
VI -sin.2  A* 

Vf-'cos.2  A 
cos.  A 

2  tang.  JA 
1  — tang.2  JA' 

2  cot.  JA 


cot.  J  A  -tang. 


cot.  A— 2  cot. 


1  —  cos.  2  A 


sin.  2A 


sin.  2A 

1  +  cos.  2A' 


v 


1  — cos.  2A 
1  +  cos.  2A* 


TRIGNOMETRICAL    FORMULAE.  353 

Relative  to  two  Arcs,  A  and  B. 

sin.  (A+B)—  sin.  A  cos.  B+cos.  A  sin.  B, 
sin.  (A  —  B)  =  sin.  A  cos.  B—  cos.  A  sin.  B, 
cos.  (A+B):=cos.  A  cos.  B  —  sin.  A  sin.  B, 
cos.  (A—  B)  =  cos.  A  cos.  B  +  sin.  A  sin.  B, 


= 
1—  tang.  A.  tang.  B 

tang.  A  —  tang.  B 
= 


sin.  (A  +  B)__tang.  A  +  tang.  B_cot.  B  +  cot.  A 
sin.  (A—  B)     tang.  A—  tang.  B     cot.  B  —  cot.  A5 
cos.  (A+B)_cot.  B  —  tang.  A_cot.  A  —  tang.  B 
cos.  (A—  B)     cot.  B  +  tang.  A     cot.  A  4-  tang.  B' 
sin.  A  cos.  B  =  i  sin.  (A+B)  +  J  sin.  (A—  B), 
cos.  A  sin.  B  =  J  sin.  (A+B)  —  £  sin.  (A—  B), 
sin.  A  sin.  B  =  J  cos.  (A—  B)  —  J  cos.  (A+B), 
cos.  A  cos.  B  =  J  cos.  (A  +  B)  +  i  cos.  (A—  B), 
sin.  A+sin.  E=2  sin.  i(A+B)  .  cos.  i(A-B), 
cos.  A+cos.  B=2  cos.  i(A  +  B)  .  cos.  J(A-B), 
sin.  A-sin.  B  =  2  sin.  i(A-B)  .  cos.  i(A-fB), 
cos.  B  —  cos.  A—  2  sin.  i(A—  B)  .  sin. 

sin.  (A  +  B) 

tang.  A  +  tang.  Brrz  -  J  --  ^, 
cos.  A  .  cos.  B 

,  sin.  (A  +  B) 

COt.  A  +  COt.  B^:-:  -  r*  -  :  -  '-, 

sin.  A  .  sin.  B 

„      sin.  (A-B) 

tang.  A-  tang.  B  =  -  ^  ---  ^, 
cos.  A  .  cos.  B 

,    _  sin.  (A—  B) 

COt.  B  —  COt.  A=^  -  ?  --  :  -  '=, 

sin.  A  .  sin.  B 

sin.  A  +  sin.  B_tang.  ^(A+B) 
sin.  A-sin.  B  ~  tang.  i(A-B)' 

cos.  B  +  cos.  A_  cot.  j(A+B) 
cos.  B—  cos.  A  "tang.  J(A—  B)' 
Z 


354  PRACTICAL   ASTRONOMY. 

Multiple  Arcs. 

sin.  2A—  2  sin.  A  cos.  A, 
sin.  3A=2  sin.  2A.  cos.  A—  sin.  A, 
sin.  4  A—  2  sin.  SA.cos.  A  —  sin.  2A, 
cos.  2A=2  cos.  A  cos.  A—  1, 

=  1-2  sin.2  A, 

cos.  3A  =  2  cos.  2A  cos.  A—  cos.  A, 
cos.  4  A  =2  cos.  3  A  cos.  A—  cos.  2  A, 

0  A        2  tang.  A 
tang.  2A=T—  -  5—., 
1  —  tang.2  A 


tang.  3A=          L, 
1-tang.  2A  tang.  A 


tana.  4A= 


tang>  3A+tang-  A 


. 
1—  tang.  3A  tang.  A" 


Trigonometrical  Series. 

.      .       .       A3  A5  A7 

~^3  +  2. 3. 4. 5~2. 3.4.5.6.7  +  '  6 

A2  ,      A4  A6 

cos.  A=l — —+.——---— ———.+  9  etc., 

A3     2A5       17A7 
tang.  A^A+-g-+g-^  +  32   g   y  +  ,  etc., 

=!--— —     2A5      etc 

A     3     3.5    3.5.7 


Differentials  of  Trigonometrical  Lines. 

d  sin.  #—  +cos.  re  c?a;, 
cf  cos.  x=  — sin.  x  dx, 

dtang.  x—  + — _, 
cos."  x 

-,                      dx 
d  cot.  x= -— . 


TABLES. 


TABLE   I. 


357 


POSITIONS  OF  THE  PRINCIPAL  FOREIGN  OBSERVATORIES. 

North  Latitudes  and  West  Longitudes  are  indicated  by  the  sign  -f  ;  South  Latitudes  and  East  Longitudes 

by  the  sign  — . 


Place. 

Latitude. 

Longitude 
from  Washington 
in  Time. 

Longitude 
from  Greenwich 
in  Time. 

Longitude 
from  Greenwich 
in  Arc. 

Abo  

+60  26  56.8 

h.    m.       s. 
—  6   37  20.0 

h.    TO.        *. 

—   i   29     8.8 

—    22    17    H.4 

Altona 

+53  32  45.3 

—  5  47  57.4 

—  o  39  46.2 

—      9    56    32.2 

Armagh 

-j-54  21    12.7 

—  4  4i  35.7 

+  o  26  35.5 

+     6  38  52.5 

Athens         .       

+37  58  20.0 

—  6  43     6.4 

—   i   34  55.2 

—  23  43  47.8 

Berlin 

+52  3o  16.7 

—6     i  46.i 

—  o  53  34.9 

—  i3  23  43.9 

Bilk 

-j-5l     12    25.0 

—  5  35  16.1 

—    O    27       4.0 

—     6  46   i  3.  9 

Bonn 

+5o  43  45.o 

—  5  36  35.7 

—  o  28  24.5 

—     7     6     6.0 

Breslau 

+5i     6  56.o 

—  6  16  21.2 

—   i     8  10  o 

/           v  •  v 
—   17     2  3o  o 

Brussels 

+5o  5i   10.7 

—  5  25  38.8 

—    O    17    27.6 

—       4    21     54.0 

Cambridge,  England. 

Cape  of  Good  Hope.  . 
Christiania 

-j-52  12  5i.8 

—33  56     3.o 

+5o  54  43  7 

—58  34.7 

—    6    22       7.2 

—  5  5i     60 

—  oo  23.5 

—   i   i3  56.o 
—  o  42  54  8 

—     o     5  53.i 

—  18  28  59.7 
—  10  43  4i  4 

Copenhagen 

-j-55  4o  53.o 

—  5  58  3o  5 

—  o  5o   10  3 

—  12  34  49  5 

Cracow       1       ... 

+5o     3  5o.o 

—  6  28     24 

—   i    19  5  i   2 

—  IQ  57  48  6 

Dorpat     

+58  22  47.1 

—  6  55     5.8 

—   i  46  54.6 

—  26  43  38.4 

Dublin  

4-53  23   i3.o 

442    49  •  2 

+    O    25    22.  O 

+     6  20  3o.o 

Durham 

-f-54  46     6.4 

—5     i   53  2 

+  o     6  18  o 

+     i   34  3o  o 

Edinburgh 

+55  57  23.2 

—  4  55  28.2 

+    O    12    43    O 

+     3   10  45  o 

Florence 

+43  46  4o  8 

—  5  53   12  9 

o  45     17 

—    II    1  5    24   Q 

Geneva 

+46  ii  58.8 

—  5  32  48  9 

O    24    37     7 

—     6     o  °  5  2 

Gbttingen  

+5i   3i  47.9 

—  5  47  57.3 

—  o  39  46  .  i 

—     9  56  3i.5 

Gotha 

+5o  56     5.2 

—  5  5i     69 

—  o  42  55  7 

—  10  43  54.9 

Greenwich 

+5i  28  38.2 

—  5     8112 

o     o     o 

o     o     o 

Hamburgh 

+53  33     5.o 

—  5  '48     4.8 

—  o  39  53  6 

—     9  58  23.4 

Kasan 

+55  4y  23   i 

8  24  43  i 

3   16  3i   o 

—  /o     7  58  o 

Kbnigsberg 

+54  42  5o.4 

—  6  3o   ii   6 

I     22        O    4 

**y      /   ^^'V 
—  20  3o     54 

Kremsmunster 

+48     3  23  8 

6     4  44  6 

o  56  33  4 

i4     8  21   3 

Leipsic 

+  5  1  20  20  .  4 

—  5  57  3o  7 

—  o  4g  28  5 

1222        7     5 

Leyden 

+  52       Q    28.2 

—  5  26     8.6 

O     17    57    A 

4    2O    21     4 

Tjjvprpool 

4-53  2A  47  8 

4  56  ii    i 

_j_           3          0           T       rj 

Madras 

+  l3       4       Q.2 

—  10  29     8.2 

—  5  20  57  o 

—  80  i4  i5  o 

Mannheim 

+4o  20  12.  o 

—  5  42     2.7 

—  o  33  5i.5 

—     8  27  52.5 

Markree  

+54  10  31.7 

—  4  34  22.8 

+  o  33  48.4 

+     8  27     6.0 

Marseilles  

+43  17  49*0 

—  5  29  4o.2 

—    O    21     29  .O 

—     5  22  i4.8 

Milan 

+45  28    0.7 

—  5  44  57.8 

—  o  36  46  6 

—     o  ii   39  .  6 

Modena 

+44  38  52.8 

—  5  5i  55.2 

—  o  43  44  o 

—   10  55  59.5 

Moscow 

+55  45  19.8 

—  7  38  28.5 

—  2  3o  17.3 

—  37  34  ig.3 

Munich 

+48     8  45  o 

—  5  54  37  6 

o  46  26  4 

ii   36  36  6 

Naples  

+4o  5  1  46.6 

—  6     5  12  .  i 

—  o  57     o.o 

—   i4  i5   13.9 

Olmutz 

+49  35  4o.o 

—  6  17  ii  .3 

—   i     o     o   i 

—   17   i5     i.5 

Oxford  

+5i  45  36.o 

—5     3     8.6 

+  o     5     2.6 

+     i   i  5  39.0 

Padua 

+45  24     2.5 

—  5  55  4o  2 

—  o  4?  20  o 

—  ii  52  i5.4 

Palermo 

+  38     6  44.o 

—6     i   36.7 

—  o  53  25.5 

—   i3  21   21.9 

Paramatta 

33  48  49  8 

4-  8  47  42  6 

•  •-  1  o     4     6  " 

—  i5i     i   33.7 

Paris 

+48  5o  1  3  2 

5   17  32  7 

•  •   o     o  °  i   5 

—       2    2O    21  .9 

Petersburgh 

_|_5o  56  20  7 

—    7       O    2,A    1 

2     i    i3.5 

—    3o    l8    22.2 

Prague 

+5o     5   18  5 

—6     5  53  2 

—  o  57  42  .  o 

—  i4  25  29.4 

Pulkowa    

-i-5o  A6  18  7 

—    7       Q    2O    Q 

—    2        I     18.7 

—  3o  19  4o.i 

Rome  

+4i   53  54.o 

—  5  58     5.Q 

—  o  4o  54.7 

—  12  28  4o.5 

San  Fernando     . 

+36  27  45.o 

4    43     22  .  I 

+o  24  4o«  i 

+     6  12   17.1 

Santiago  

—33  26  24.8 

—  o  25  52.3 

+  4  42  18.9 

+  70  34  43.5 

Senftenberg  

+  5o     5  10.  i 

—  6   i4     i  •  i 

—  i     5  49.9 

—   16  27  28.9 

Vienna 

+48  12  35.5 

—  6  i3  43.7 

—i     5  32.5 

—   16  23     7.9 

Wilna 

+54  4o  59.i 

—  6  49  23.o 

—  i  4i   ii.  8 

—  25   17  56.5 

358 


TABLE    II. 


LATITUDES  AND  LONGITUDES  OF  PLACES  IN  THE  UNITED  STATES. 

West  Longitudes  are  considered  as  positive,  East  Longitudes  as  negative. 


Place. 

Latitude. 

Longitude 
from  W  ashington 
in  Time. 

Longitude 
from  Greenwich 
in  Time. 

Longitude 
from  Greenwich 
in  Arc. 

Burlington,  Vt.  ...  .  .  _  . 

44    27 

h.  m.     s. 
—  o  i5  3i 

h.   m.     s. 

4  52  4o 

73    10 

Middlebury,  Vt 

44    o 

—  o  i5  3g 

4    52     32 

73     8 

Brunswick,  Me.  ,  College  
Hanover,  N.  H.  

43  53     o 
43  42 

—  o  28  3i 

—  O    IQ    IO 

4  39  4o.i 
4  48  52 

69  55     i 

72    l3 

Rochester,  N.  Y. 

43     8  17 

-f-o     3   i3 

5  ii   24 

77    5  1 

Clinton,  N.  Y  

43     2 

—  o     6  i5 

5     i   56 

75    2Q 

Schenectady,  N  Y. 

42  48 

O    12    3l 

4  55  4o 

73  55 

Williamstown,  Mass. 

42  42  49 

—  o  i5   18  6 

4  52  52  6 

y3     j3     I0 

Albany,  N.  Y.,  Capitol 

42  39     3 

—  o  i  3   ii   o 

4  54  59  3 

73  AA  Aa 

Beloit,  Wis  

42    32 

4-0  47  53 

5  56    4 

80     i 

Cambridge,  Mass.,  Observat. 
Amherst,  Mass.,  College  
Boston,  Mass.,  State  House. 
Ann  Arbor,  Mich. 

42    22    48.6 
42    22    l5.6 
42    21     27.6 
42     17 

—  o  23  4i.5 
—  o  i  8     5.2 
—  o  23  57 

+  O    27        I 

4  44  29.7 
4  5o     6 
4  44  i4 
5  35  12 

71      7  24.9 
72  3i   28 
71      3   3o.o 
83  48 

Chicago,  111. 

4l     52    2O 

+  O    42        Q 

5  5o  20 

87  35 

Providence,  R.  L,  College.. 
Hartford,  Conn.,  State  House 
Middletown,  Conn.,  College. 
West  Point,  N.  Y. 

4i  5o  16.7 
4i  45  59 
4i  33     8 
4i  23  a5.6 

—  o  22  36.6 
—  o   17  28.2 
—  o  17  35 

O     12     23     I 

4  45  34.6 
4  5o  43 
4  5o  36 
4  55  48  i 

71   23  39.7 
72  4o  45 
72   39     o 

73     57        10 

New  Haven,  Conn.,  Coll.  Sp. 

Nantucket,  Mass.,  Mite.  Obs. 
Hudson,  Ohio,  Observatory  . 
New  York,  City  Hall  .  .. 

4i   18  27.7 

4i    16  57.2 
4i  i4  42.6 
4o  42  43.2 

—  o   16  29.6 
—  o  27  48.6 

-f-O    17    32.1 
—  O    12     II     O 

4  5i  4i.6 

4  4o  22.6 
5  25  43.3 
4  56     o  .  2 

/  ^    •*¥       l  •  u 

72   55  24.1 

70     5  39.0 
81   25  48.9 

nA        o        3     I 

Princeton,  N.  J.,  College... 
Canonsburgh,  Penn. 

40    20    52.1 
Ao    17 

—  o     9  34.2 
-f-o  i3     5 

4  58  37.o 
5  21    1  6 

74  39  i5.3 

Carlisle,  Penn  

4o    12 

-I—  O       O    4l 

5     8  52 

77     1  3 

Crawfordsville,  la 

4o    3 

-4-o  3o      5 

5  An  16 

/  /     *  ^ 

86  49 

Philadelphia,  High  Sch.  Obs. 
Jacksonville,  111. 

39  57     7.5 
3o  45 

—  o     7  33.6 
-f-o  53     i 

5     o  37.6 

6        112 

75     9  23.4 

Oxford,  Ohio 

39  3o 

-fo  3o  53 

5  3o     A 

84  46 

Athens,  Ohio 

3o  21 

5  28  28 

82     7 

Baltimore,  Wash.  Monument 
Bloomington,  la  

39   17  47  ..8 

89     1  2 

—  o     i   44.6 
+o  37  4i 

5     6  26.6 
5  45  52 

76  36  38.6 
86  28 

Cincinnati,  Ohio,  Observatory 
Annapolis,  Md.,  State  House 
Georgetown,  D,  C.,  Observ.  . 
Washington,  D.  C.,  Observ.. 
St.  Louis,  Mo. 

39     5  54 
38  58  4o.2 
38  54  26.1 
38  53  39.3 
38  37  28 

-f-o  29  46.9 

—  O       2    l4-6 

-|-o     o     6.2 

000 

_|_o  52  49  5 

5  37  58.i 
5     5  56.6 
5     8   17.4 
5     8   ii.  2 
6     i     07 

84  29  3o.8 
76  29     9.4! 
77     421.0 
77     2  48.o 

Lexington,  Ky  

38     6 

-J-o  29      i 

5  37  12 

84   1  8 

Charlottesville,  Va.,  Univers. 

SanFrancisco,  Cal.,  San  Jose 
Monterey,  Cal.,  Observatory  . 
Nashville,  Tenn.,  University 
Chapel  Hill,  N.  C. 

38     2     3 

37  48  23.  .6 
36  37  59.9 
36     9  33 

35  54  21 

-fo     5   54.7 
+3      i   27.4 

-j-2     59    26.3 

-j-o.  39     5.,o 
-j-o     8  5o 

5  i4    5.9 

8     9  38.6 
8     7  37.5 
5  47   16.2 
5  17  10 

78  3  1.  29 

122    ^A'    39.6 
121     54    22.0 

86  49     3 

7Q     17    3o 

Santa  Fe,  New  Mexico  
Columbia,  S.  C. 

35  4i     6 
33  5? 

+  i  55  54 
+o  16   17 

7     4     5.5 
5  24  28 

IO6        I     22 

8l       7 

Athens,  Ga. 

33  54 

-f-O    25    37 

5  33  48 

83  27 

Tuscaloosa,  Ala. 

33   12 

_LO    /2    37 

5  5o  48 

87  42 

Charleston,  S.  C.,  Gibbes'  Ob. 
San  Diego,  Cal.,  Observat'y. 

Savannah,  Ga.,  Exchange  .. 
Mobile,  Ala.,  Episcopal  Spire 
New  Orleans,  City  Hall  
Galveston,Tex.,  Court-house 

32  47     5.3 
32  4i   58.o 

32     4  53.4 
3o  4i   26.2 
29  57  3o 
29   18   i4.5 

+0  ii   32.8 

+  2    40    42.3 

-(-o  16     9.9 
-fo  43  54-7 
-fo  5i  48.8 
-f-  1    10  55.o 

5  19  44  -o 
7  48  53.5 

5  24  21  .  i 
5  53     5.9 
600 
6   19     6.2 

79  55  59.8 

117     l3     22.0 

81     5   16.8 
88     i   29.2 
90     o     o 
94  46  33.8 

TABLE    III. 


359 


To  convert  Hours,  Minutes,  and  Seconds  into 
Decimals  of  a  Day. 

To  convert  Decimals  of  a  Day  into 
Hours,  Minutes,  and  Seconds.  * 

Hours 

Decimal. 

Min.  |  Decimal.    Sec. 

Decimal. 

Dec. 

H.  M,  S. 

Dec. 

H.  M.   S. 

I 

.0416+ 

! 

.000694+ 

i 

.OOOOI  I  6 

.01 

o  i4  24 

.6! 

i4  38  24 

2 

.o833+ 

2 

.ooi388+ 

2 

.OOOO23l 

.  02 

o  28  48 

.62 

i4  52  48 

3 

.  i25o+ 

3 

.002083  + 

3 

.OOOO347 

.08 

o  43  12 

.63 

i5  7  12 

4 

.1666-4- 

4 

.002777+ 

4 

.oooo463 

.04 

o  57  36 

.64 

i5  21  36 

5 

.2083+ 

5 

.oo3472+ 

5 

.0000579 

.o5 

I  12   O 

.65 

i5  36  o 

6 

,25oo+ 

6 

.004166+ 

6 

.0000694 

.06 

I  26  24 

.66 

i  5  5o  24 

7 

.ao  1  64* 

7 

.004861  + 

7 

.0000810 

.07 

i  4o  48 

.67 

'16  4  48 

8 

.3333+ 

8 

.oo5555+ 

8 

.0000925 

.08 

i  55  12 

.68 

16  19  12 

9 

.375o+ 

9 

,oo625o  + 

9 

.0001042 

.09 

2  9  36 

.69 

16  33  36 

10 

.4i66+ 

10 

.006944+ 

10 

.0001157 

.  10 

2  24   0 

.70 

16  48  o 

ii 

.4583+ 

ii 

.007638  + 

ii 

.0001278 

.  ii 

2  38  24 

•71 

17   2  24 

12 

.6000+ 

12 

.oo8333+ 

12 

.0001389 

.  12 

2  52  48 

.72 

17  i  6  48 

i3 

.54i6+ 

i3 

.009027+ 

i3 

.0001  5o5 

.i3 

3  7  12 

.73 

17  3i  12 

i4 

.5833+ 

i4 

.009722+ 

i4 

.0001620 

.14 

3  21  36 

•74 

17  45  36 

i5 

.625o+ 

i5 

.010416+ 

i5 

.0001736 

.i5 

3  36  o 

.75 

1800 

16 

.6666+ 

16 

.OIIII  1  + 

16 

.oooi852 

.16 

3  5o  24 

.76 

18  i4  24 

!? 

.7o83+ 

i? 

.on8o5  + 

I7 

.0001968 

•ll 

4  4  48 

•77 

18  28  48 

18 

•  75oo+ 

18 

.oi25oo+ 

18 

.0002083 

.18 

4  19  12 

.78 

18  43  12 

*9 

.7916+ 

J9 

.018194+ 

J9 

.0002199 

.19 

4  33  36 

•79 

18  57  36 

20 

.8333+ 

20 

.oi3888+ 

20 

.ooo23i5 

.20 

4  48  o 

.80 

19  12   O 

21 

.875o+ 

21 

.oi4583+ 

21 

.ooo243i 

.21 

5   2  24 

.81 

19  26  24 

22 

.9166+ 

22 

.015277+ 

22 

.0002546 

.22 

5  16  48 

.82 

19  4o  48 

23 

.9583+ 

23 

.015972+ 

23 

.0002662 

.23 

5  3i  12 

.83 

19  55  12 

24 

I  .0000  + 

24 

.016666+ 

24 

.0002778 

.24 

5  45  36 

.84 

20  9  36 

25 

.017361  + 

25 

.0002894 

.  25 

600 

.85 

20  24  o 

26 

.oi8o55  + 

26 

.0003009 

.26 

6  i4  24 

.86 

20  38  24 

CD 

27 

.018750+ 

27 

.ooo3i25 

.27 

6  28  48 

.87 

20  52  48 

J-. 

i 

28 

.019444+ 

28 

.ooo324i 

o 

.  20 

6  43  12 

.88 

21   7  12 

i 

29 

.020138  + 

29 

.ooo3356 

.29 

6  57  36 

.89 

21  21  36 

1 

3o 

.020833+ 

3o 

.0003472 

.80 

7120 

.90 

21  36  o 

43 

3i 

.021527+ 

3i 

.ooo3588 

.81 

7  26  24 

.91 

21  5o  24 

| 

32 

.022222  + 

32 

.0008704 

.32 

7  4o  48 

.92 

22  4  48 

^D 

-|j 

33 

.022916+ 

33 

.0003819 

.33 

7  55  12 

.93 

22  19  12 

34 

.0236n  + 

34 

.0008935 

.34 

8  9  36 

•  94 

22  33  36 

73 

35 

.o243o5+ 

35 

.ooo4o5i 

.35 

8  24  o 

.96 

22  48   0 

CD 

<c! 

36 

.O25OOO  + 

36 

.0004167 

.36 

8  38  24 

.96 

23   2  24 

a 

37 

.025694+ 

37 

.0004282 

.87 

8  52  48 

•97 

23  16  48 

CUD 
'w 

38 

.026388+ 

38 

.0004398 

.38 

9  7  12 

.98 

23  3l  12 

of  j/ 

39 

.027083+ 

39 

,ooo45i4 

.89 

9  21  36 

•99 

23  45  36 

If 

4o 

.027777+ 

4o 

.0004680 

.4o 

9  36  o 

/ 

-M  G 
03  ^5 

4i 

.028472+ 

Ii 

.ooo4745 

•  4i 

9  5o  24 

•  OO  I 

O   I  2  U  .  /J. 

r>   £0   Q 

)5  "" 

42 

.029166+ 

42 

.0004861 

.42 

10  4  48 

•  OO2 

r\r\^ 

Jt   J  J,  •  O 

S5 

43 

.029861  + 

43 

.0004977 

.43 

IO  19  12 

•  ooo 

4  i9»2 
^  /K  & 

fil 

44 

.o3o555  + 

44 

.oooSogS 

•  44 

10  33  36 

•  OO^. 

O  /I  J  •  U 

«  s. 

45 

,o3i25o+ 

45 

,ooo52o8 

.45 

10  48  o 

•  oo£) 

*7  I  2  •  O 
Q  QQ   / 

It 

46 

.o3i944+ 

46 

.ooo5324 

.46 

II   2  24 

•  OOD 

O  OO  .  /f 
rn   /  8 

3 
gj 

47 

.032638+ 

47 

.ooo544o 

•47 

ii  16  48 

.  007 

nnQ 

IO   Z|.  .  O 

T  T   ^  T   O 

O 

48 

.033333+ 

48 

.ooo5556 

.48 

II  31  12 

•  OOO 

II  O  I  .  £ 

ip 

r3 

49 

.034027+ 

49 

.0005671 

•  49 

ii  45  36 

•  009 

12  57.6 

I 

5o 

.034722  + 

5o 

.0005787 

.5o 

12   0   0 

•  OI  O 

I  L\  24  .  O 

9 

CD 

5i 

.o354i6+ 

5i 

.0005908 

.5i 

12  l4  24 

.0001 

o  o  8  .  64 

& 

52 

.o36in  + 

52 

.0006019 

.52 

12  28  48 

.0002 

o  17,28 

rt 

53 

.o368o5+ 

53 

.0006  i  34 

.53 

12  43  12 

.ooo3 

o  25.92 

+ 

54 

.037500+ 

54 

.ooo625o 

.54 

12  57  36 

.ooo4 

o  34.56 

a 

55 

.038194+ 

55 

.ooo6366 

.55 

l3  12   O 

.ooo5 

0  43.20 

tD 

*8 

56 

.038888+ 

56 

.0006481 

.56 

1  3  26  24 

.0006 

o  5i.84 

S 

57 

.039683+ 

57 

.0006597 

.57 

18  4o  48 

.0007 

i  o.48 

H 

58 

.040277+ 

58 

.0006718 

.58 

i3  55  12 

.0008 

i  9.12 

59 

.040972+ 

59 

.0006829 

.59 

i4  9  36 

.0009 

i  17.76 

60 

.041666+ 

60 

.0006944 

.60 

i4  24  o 

.0010 

i  26.40 

360 


TABLE    IY. 


To  convert  intervals  of  Mean  Solar  Time  into  equivalent  intervals  of  Sidereal  Time. 


HOURS. 

MINUTES. 

SECONDS.                    ||                FRACTIONS. 

Mean  T.     Sidereal  Time. 

Mean  T.   Sidereal  Time 

MeanT 

Sidereal  Time.  Mean  T. 

Sidereal  Time. 

h.      ,  h.  m.          s. 

7/1. 

m.       s. 

s. 

s. 

s. 

s. 

1 

I     0       9.866 

I 

i   o.i  64 

I 

i.oo3 

0.02 

O.O2O 

2 

2    0     19.713 

2 

2    O.329 

2 

2.006 

o.o4 

o.o4o 

3 

3  o  29.669 

3 

3  0.493 

3 

3.oo8 

0.06 

O.o6o 

4 

4  o  39.426 

4 

4  0.667 

4 

4.  on 

0.08 

O.oSo 

5 

5  o  49.282 

5 

5  0.821 

5 

$.01,4 

0.  10 

0.  100 

6 

6  o  59.i39 

6 

6  0.986 

6 

6.016 

0.  12 

0.  120 

7 

7  i     8.996 

7 

7  i.i5o 

7 

7,019 

0.  l4 

o.  i4o 

8 

8  i   18.862 

8 

8     .3i4 

8 

8.022 

0.16 

o.  1  60 

9 

9  i   28  .  708 

9 

9     .478 

9 

9.025 

0.18 

0.180 

10 

10  i    38.565 

10 

10      .643 

10 

10.027 

0.20 

0.201 

ii 

ii    i   48.421 

ii 

ii      .807 

ii 

ii  .o3o 

0.22 

0.221 

12 

12     I     68.278 

12 

12        .971 

12 

I2.o33 

O.24 

0.24l 

i3 

i3  2     8.i34 

1.3 

i3  2.i36 

i3 

i3.o36 

O.26 

0.26l 

i4 

i4  2  17.991 

i4 

i4  2.3oo 

i4 

i4.o38 

0.28 

0.28l 

i5 

16  2  27.847 

i5 

i5  2.464 

16 

i5.o4i 

o.3o 

0.36| 

16 

16  2  37.704 

16 

i  6  2.628 

16 

i  6  .  o44 

0.32 

0.321 

J7 

17    2    47.56o 

17 

17  2.793 

17 

17.047 

o.34 

o.34i 

18 

18  2  67.417 

18 

18  2.967 

18 

18.049 

o.36 

o.36i 

T9 

19  3     7.278 

*9 

19  3.  121 

T9 

19.062 

o.38 

o.38i 

20 

20  3   17.  129 

20 

20    3.285 

20 

20.066 

o.4o 

o.4oi 

21 

21   3  26.986 

21 

21     3.450 

21 

21  .067 

0.42 

0.421 

22 

22  3   36.842 

22 

22     3.6l4 

22 

22.060 

0.44 

0.441 

23 

23  3  46.699 

23 

23  3.778 

23 

23.o63 

0.46 

0.461 

24 

24  3  56.555 

24 

24  3.943 

24 

24.066 

0.48 

o.48  1 

25 

26    4-  IO7 

26 

26.068 

o.5o 

o.5oi 

26 

26    4.271 

26 

26.071 

0.62 

0.621 

27 

27    4.435 

27 

27.074 

o.54 

o.54i 

®    0 

28 

28    4.600 

28 

28.077 

0.66 

0.662 

•£    § 

29 

29    4.764 

29 

29.079 

o.58 

0.682 

*3    & 

3o 

3o  4.928 

3o 

3o.o82 

0.60 

0.602 

II 

3i 

3i   6.093 

3i 

3i.o85 

0.62 

0.622 

3  s 

32 

32  6.267 

32 

32.  088 

o.64 

0.642 

'*   bo 

33 

33  6.421 

33 

33.090 

0.66 

0.662 

o  .S 

•g'Q   • 

34 

34  5.585 

34 

34.093 

0.68 

0.682 

'Si  I 

35 

35  6.760 

35 

35.096 

0.70 

0.702 

111 

36 

36  5.9i4 

36 

36.099 

0.72 

0.722 

w         rt 

<D     ff> 

37 

37  6.078 

37 

37.  ioi 

0.74 

0.742 

e-  2  S     . 
5'S  Ben 

38 

38  6.242 

38 

38.io4 

0.76 

0.762 

CD                     £H     C* 

S~s  S  rH 

39 

3g  6.407 

39 

Sg.  107 

0.78 

0.782 

^        >            • 
<+-<                  '^        PU 

4o 

4o  6.671 

4o 

4o.  no 

0.80 

0.802 

o  »  be 

f-     S     ni     Of 

4i 

4i   6.735 

4i 

4i  .  112 

0.82 

0.822 

5  .a  2r3 
,g  •**  .fij^i, 

42 

42  6.900 

42 

42.116 

o.84 

0.842 

M  'Tj     Q     S 

43 

43  7.064 

43 

43.ii8 

0.86 

0.862 

mi 

44 

44  7.228 

44 

44.  120 

0.88 

0.882 

BJ^ 

45 

45  7.392 

45 

45.123 

o.9o 

0.9O2 

|!1| 

46 

46  7.667 

46 

46.  126 

0.92 

o.923 

•5  B'i 

47 

47  7-721 

47 

47.129 

o.94 

o.943 

g'E  S1 

48 

48   7.885 

48 

48.i3i 

o.96 

o.963 

s-y 

49 

49  8.049 

49 

49.i34 

o.98 

o.983 

M 

5o 

r 

5o  8.214 

5o 

60.137 

c           / 

I  .00 

i.ooS 

3       QJ 

61 

5i  8.378 

5i 

5i  .  i4o 

.2  | 

62 

62  8.542 

52 

62.  142 

J2   '"** 

53 

53  8.707 

53 

53.i45 

H   "^ 

54 

54  8.871 

54 

54.i48 

M      QJ 

55 

55  9.o35 

55 

55.i5i 

'-3:2 

56 

56  9.199 

56 

56.i53 

H02 

57 

57  9.  364 

57 

67.166 

58 

58  9.628 

58 

58.i59 

59 

69  9.692 

59 

59.  162 

TABLE   V. 


361 


To  convert  intervals  of  Sidereal  Time  into  equivalent  intervals  of  Mean  Solar  Time. 


HOURS. 

MINUTES. 

SECONDS. 

FRACTIONS. 

Sider.  T.j       Mean  Time. 

Sider.  T. 

Mean  Time. 

Sider.  T. 

Mean  Time. 

Sider.  T. 

Mean  Time. 

h. 

A.     771.                S. 

»t. 

771.              S. 

s. 

s. 

s. 

s. 

I 

o  59  5o.  170 

I 

o  59.836 

I 

0-997 

0.02 

0.020 

2 

i   Sg  4o.34i 

2 

i   59.672 

2 

1.995 

o.o4 

o.o4o 

3 

2  59   3o.5n 

3 

2  59  .  509 

3 

2.992 

O.O6 

O.060 

4 

3  Sg   20.682 

4 

3  59.345 

4 

3.989 

0.08 

O.o8o 

5 

4  59   io.852 

5 

4  59.181 

5 

4.986 

O.  IO 

O.  100 

6 

5  59     1.023 

6 

5  59.017 

6 

5.984 

0.  12 

O.  120 

7 

6  58   Si.igS 

7 

6  58.853 

7 

6.981 

0.  l4 

o.  i4o 

8 

7  58  4i.364 

8 

7  58.  689 

8 

7.978 

0.16 

o.  160 

9 

8  58   3i.534 

9 

8  58.526 

9 

8.975 

0.18 

o.  180 

10 

9  58  21.  70^ 

10 

9  58.362 

10 

9-973 

0.2O 

0.199 

ii 

10  58   ii  .875 

ii 

10  58.198 

ii 

io.97o 

O.22 

0.219 

12 

ii   58     2.o45 

12 

ii  58.o34 

12 

11.967 

O.24 

0.239 

i3 

12    57     52.  2l6 

i3 

12    57.870 

i3 

12.965 

O.26 

0.259 

i4 

i3  57  42.386 

i4 

i3   57.706 

i4 

13.962 

0.28 

0.279 

i5 

i4  57   32.557 

i5 

i4  57.543 

i5 

14.959 

o.3o 

0.299 

16 

l5    57     22.727 

16 

i5  57.379 

16 

i5.956 

0.32 

o.3i9 

17 

16  57   12.897 

17 

16  57.2i5 

17 

i6.954 

o.34 

o.339 

18 

17  57    3.068 

18 

i7  57.o5i 

18 

17.951 

o.36 

o.359 

19 

18  56   53.238 

19 

18  56.  887 

19 

18.948 

o.38 

o.379 

20 

19  56  43.4o9 

20 

19  56.723 

20 

19.945 

o.4o 

0.399 

21 

20  56   33.579 

21 

20  56.56o 

21 

20.943 

0.42 

0.419 

22 

21   56  23.750 

22 

21   56.396 

22 

21  .940 

o.44 

0.439 

23 

22  56  13.920 

23 

22     56.232 

23 

22«937 

0.46 

0.459 

24 

23  56     4.091 

24 

23  56.o68 

24 

23.934 

o.48 

0.4-79 

25 

24  55.904 

25 

24.932 

o.5o 

0.499 

26 

25  55.74i 

26 

25.929 

O.52 

0.519 

27 

26  55.577 

27 

26.926 

0.54 

o.539 

JO 

28 

27  55.4i3 

28 

27-924 

o.56 

0.558 

§ 

29 

28  55.249 

29 

28.921 

o.58 

o.578 

H-a 

3o 

29  55.o85 

3o 

29.918 

0.60 

o.598 

3i 

3o  54.921 

3i 

3o  .915 

0.62 

0.618 

03     CD 

32 

3i   54-758 

32 

3i  .913 

o.64 

0.638 

S  ^35 

33 

32.  54.594 

33 

32  .  910 

0.66 

0.658 

34 

33  54.43o 

34 

33.907 

0.68 

o.678 

SIS'-*3 

35 

34  54.266 

35 

34.904 

0.70 

o.698 

C3    o  •—  * 

36 

35  54.102 

36 

35.902 

0.72 

o.7i8 

"rt   &§     . 

37 

36  53.938 

37 

36.899 

0.74 

o.738 

§  2^  S 

38 

37  53.775 

38 

37.896 

0.76 

o.758 

"55    -(-3      CM      Q} 

39 

38   53.6n 

39 

38.894 

0.78 

o.778 

4o 

39  53.447 

4o 

39.89i 

0.80 

o.798 

^  2  'So  ^ 

4i 

4o  53.283 

4i 

4o.888 

0.82 

0,818 

.2  v§  S  ^ 

42 

4i  53.119 

42 

4i.885 

o.84 

0.838 

§  i^3  s" 

43 

42  52.955 

43 

42.883 

0.86 

o.858 

c  1  ^  x 

44 

43     52.792 

44 

43.88o 

0.88 

o.878 

8     §H 

45 

44  52.628 

45 

44.877 

0.90 

0.898 

g  i*  8 

46 

45  52.464 

46 

45.874 

0.92 

0.917 

•S  o>  .£cQ 

47 

46  52.3oo 

47 

46.872 

0.94 

o.937 

i§  JL® 

48 

47  52.i36 

48 

47.869 

0.96 

o.957 

'a  ^  2 

49 

48   5i.973 

49 

48.866 

0.98 

o-977 

<So£ 

5o 

49  5  i  .809 

5o 

49-863 

i  .00 

o-997 

s  S  ' 

5i 

5o  5i  .645 

5i 

5o.86i 

en  "*"* 

52 

5i   5i.48i 

52 

5i.858 

^-§ 

53 

52   5i.3i7 

53 

52.855 

H   a 

54 

53   5i.i53 

54 

53.853 

a  3 

55 

54   50.990 

55 

54.85o 

_S  S 

56 

55  5o.826 

56 

55.847 

H^ 

57 

56  5o.662 

57 

56.844 

58 

57  50.498 

58 

57.842 

59 

58   5o.334 

59 

58.839 

362 


TABLE   VI. 


To  convert  degrees  into  Sidereal  Time. 


Arc 

Time 

Arc 

Time 

Arc.  Time. 

Arc.  Time. 

Arc.  Time. 

Arc. 

Time. 

Arc.!  Time 

Arc. 

Time. 

0 

h.  m. 

0 

h.  m 

0 

h.  m 

0 

h.  m 

0 

h.  m 

0 

h.  m 

/ 

m.  s 

// 

s. 

I 

o  4 

61 

4  4 

121 

8  4 

181 

12   4 

241 

16  4 

3oi 

2O   L 

I 

o  t 

I 

0.067 

2 

o  8 

62 

4  8 

122 

8  8 

182 

12   8 

242 

16  8 

302 

20  8 

2 

o  8 

2 

o.i33 

3 

O  12 

63 

4  12 

123 

8  12 

i83 

12  12 

243 

16  12 

3o3 

2O  12 

3 

0  12 

3 

0.20O 

4 

o  16 

64 

4  16 

124 

8  16 

1  84 

12  l6 

244 

16  16 

3o4 

20  1  6 

4 

o  16 

4 

0.267 

5 

O  20 

65 

4  20 

125 

8  20 

i85 

12  2O 

245 

16  20 

3o5 

20  20 

5 

o  20 

5 

0.333 

6 

0  24 

66 

4  24 

126 

8  24 

186 

12  21 

246 

1  6  2; 

3o6 

2O  24 

6 

O  22 

6 

o.4oo 

7 

o  28 

67 

4  28 

127 

8  28 

187 

12  28 

247 

16  28 

3o7 

2O  28 

7 

o  28 

7 

o.467 

8 

0  32 

68 

4  32 

128 

8  32 

188 

12  32 

248 

16  32 

3o8 

20  32 

8 

0  32 

8 

0.533 

9 

o  36 

69 

4  36 

129 

8  36 

189 

12  36 

249 

16  36 

309 

20  36 

9 

o  36 

9 

0.600 

10 

o  4o 

70 

4  4o 

r3o 

8  4o 

190 

12  4o 

250 

16  4o 

3io 

20  4o 

IO 

o  4o 

10 

o.667 

ii 

o  44 

71 

4  44 

i3i 

8  44 

i9i 

12  44 

25l 

16  44 

3u 

20  44 

ii 

o  44 

ii 

o.733 

12 

o  48 

72 

4  48 

132 

8  48 

I92 

12  48 

252 

16  48 

3l2 

20  48 

12 

o  48 

12 

0.800 

i3 

0  52 

73 

4  52 

i33 

8  52 

i93 

12  52 

253 

1  6  52 

3i3 

20  52 

i3 

0  52 

i3 

o.867 

i4 

o  56 

74 

4  56 

1  34 

8  56 

i94 

12  56 

254 

16  56 

3i4 

20  56 

i4 

o  56 

i4 

o.933 

i5 

o 

75 

5  o 

r35 

9  ° 

i95 

i3   o 

255 

17  o 

3i5 

21   O 

i5 

o 

i5 

i  .000 

16 

4 

76 

5  4 

i36 

9  4 

i96 

i3  4 

256 

17  4 

3i6 

21   4 

16 

L 

16 

i.o67 

*7 

8 

77 

5  8 

i37 

9  8 

i97 

i3  8 

257 

17  8 

3i7 

21   & 

J7 

8 

17 

1.183 

18 

12 

78 

5  12 

i38 

9  12 

198 

13   12 

258 

17  12 

3i8 

21  12 

18 

12 

18 

i  .200 

J9 

16 

79 

5  16 

i39 

9  16 

i99 

i3  16 

259 

17  16 

3i9 

21  16 

J9 

16 

J9 

i  .26-7 

20 

20 

80 

5  20 

i4o 

9  20 

200 

l3  20 

260 

17  20 

320 

21  20 

20 

20 

20 

1.333 

21 

24 

81 

5  24 

i4i 

9  24 

201 

1  3  24 

261 

17  22 

321 

21  22 

21 

•2.L 

21 

i  .4oo 

22 

28 

82 

5  28 

142 

9  28 

202 

i3  28 

262 

17  28 

322 

21  28 

22 

28 

22 

1.4*7 

23 

32 

83 

5  32 

i43 

9  32 

203 

i3  32 

263 

17  3s 

323 

21  32 

23 

32 

23 

1.533 

24 

36 

84 

5  36 

1  44 

9  36 

204 

i3  36 

264 

17  36 

322 

21  36 

24 

36 

24 

i  .600 

25 

4o 

85 

5  4o 

i45 

9  4o 

2O5 

i3  4o 

265 

17  4o 

325 

21  40 

25 

4o 

25 

1.667 

26 

44 

86 

5  44 

i46 

9  44 

206 

i3  44 

266 

17  44 

326 

21  44 

26 

44 

26 

i.733 

27 

48 

87 

5  48 

i47 

9  48 

207 

1  3  4& 

267 

17  48 

327 

21  48 

27 

48 

27 

1.800 

28 

52 

88 

5  52 

148 

9  52 

208 

i3  5s 

268117  52 

328 

21  52 

28 

52 

28 

1.867 

29 

56 

89 

5  56 

r49 

9  56 

2O9 

i3  56 

269  17  56 

329 

21  56 

29 

56 

29 

i.933 

3o 

2   0 

9° 

6  o 

i5o 

10  o 

210 

i4  c 

270  18  o 

33o 

22   0 

3o 

2   0 

3o 

2.000 

3i 

2   4 

91 

6  4 

i5i 

10  4 

211 

i4  4 

271  18  4 

33i 

22   L 

3i 

2   4 

3i 

2.o67 

32 

2   8 

92 

6  8 

l52 

10  8 

212 

14  8 

272 

18  8 

332 

22   8 

32 

2   8 

32 

2.i33 

33 

2  12 

93 

6  12 

i53 

IO  12 

2l3 

i4  12 

273 

18  12 

333 

22  12 

33 

2  12 

33 

2.200 

34 

2  16 

94 

6  16 

54 

10  16 

214 

i4  16 

274 

18  16 

334 

22  l6 

34 

2  16 

34 

2.267 

35 

2  20 

95 

6  20 

i55 

10  20 

2l5 

l4  20 

275 

18  20 

335 

22  20 

35 

2  20 

35 

2.333 

36 

2  24 

96 

6  24 

i56 

10  24 

216 

l4  2Z 

276 

18  24 

336 

22  24 

36 

2  24 

36 

2.4OO 

37 

2  28 

97 

6  28 

i57 

10  28 

2I7 

14  28 

277 

18  28 

337 

22  28 

37 

2  28 

37 

2.467 

38 

2  32 

98 

6  32 

r  58 

10  3s 

218 

14  32 

278 

18  32 

338 

22  32 

38 

2  32 

38 

2.533 

39 

2  36 

99 

6  36 

i59 

10  36 

2I9 

14  36 

279 

18  36 

339 

22  36 

39 

2  36 

39 

2.600 

4o 

2  40 

100 

6  4o 

1  60 

10  4o 

22O 

14  4o 

280 

18  4o 

34o 

22  40 

4o 

2  40 

4o 

2.667 

4i 

2  44 

101 

6  44 

161 

10  44 

221 

i4  44 

281 

18  44 

34  1 

22  J\L 

4i 

2  44 

4i 

2.733 

42 

2  48 

IO2 

6  48 

162 

10  48 

222 

i4  48 

282 

18  48 

342 

22  48 

42 

2  48 

42 

2.800 

43 

2  52 

io3 

6  52 

i63 

IO  52 

223 

14  52 

283 

18  52 

343 

22  52 

43 

2  52 

43 

2.867 

44 

2  56 

io4 

6  56 

1  64 

10  56 

224 

14  56 

284 

18  56 

344 

22  56 

44 

2  56 

44 

2.933 

45 

3  o 

io5 

7  ° 

165 

II    0 

225 

i5  c 

285 

19  o 

345 

23   0 

45 

3  o 

45 

3  .000 

46 

3  4 

1  06 

7  4 

166 

ii  4 

226 

i5  4 

286 

19  4 

346 

23  4 

46 

3  4 

46 

3.o67 

4? 

3  8 

107 

7  8 

167 

ii  8 

227 

i5  8 

287 

19  8 

347 

23  8 

47 

3  8 

47 

3.i33 

48 

3  12 

1  08 

7  12 

168 

II  12 

2a8l5  12 

288 

19  12 

348 

23  12 

48 

3  12 

48 

3.  200 

49 

3  16 

109 

7  16 

169 

ii  16 

229 

i5  16 

289 

19  16 

349 

23  1  6 

49 

3  16 

49 

3.267 

5o 

3  20 

no 

7  20 

170 

II  20 

230 

l5  20 

290 

19  20 

35o 

23  20 

5o 

3  20 

5o 

3.333 

5i 

3  24 

in 

7  24 

171 

II  24 

23l 

i5  24 

291 

19  24 

35i 

23  24 

5i 

3  24 

5i 

3.4oo 

52 

3  28 

I  12 

7  28 

172 

II  28 

232 

i5  28 

292 

19  28 

352 

23  28 

52 

3  28 

52 

3.467 

53 

3  32 

n3 

7  32 

i73 

II  32 

233 

i5  32 

293 

19  32 

353 

23  32 

53 

3  32 

53 

3.533 

54 

3  36 

n4 

7  36 

i74 

ii  36 

234 

i5  36 

294 

19  36 

354 

23  36 

54 

3  36 

54 

3.6oo 

55 

3  4o 

n5 

7  4o 

I75 

1  1  4o 

235 

i5  4o 

295 

19  4o 

355 

23  40 

55 

3  4o 

55 

3.667 

56 

3  44 

116 

7  44 

178 

ii  44 

236 

i5  44 

296 

19  44 

356 

23  44 

56 

3  44 

56 

3.733 

57 

3  48 

117 

7  48 

177 

ii  48 

237 

i5  48 

297 

19  48 

57 

23  48 

57 

3  48 

57 

3.8oo 

58 

3  52 

118 

7  52 

178 

II  52 

238 

i5  52 

298 

19  52 

58 

23  52 

58 

3  52 

58 

3.867 

59 

3  56 

119 

7  56 

179 

ii  56 

239 

i5  56 

299 

19  56 

59 

23  56 

59 

3  56 

59 

3.933 

60 

4  o 

120 

8  o 

180 

12   0|j24o 

16  o 

3oo 

20  o 

60 

24   0 

60 

4  o 

60 

4.ooo 

TABLE   VII. 


363 


To  convert  Sidereal  Time  into  Degrees. 


Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

h. 

0 

771. 

0    ' 

s. 

/  // 

*. 

" 

8. 

// 

I 

i5 

I 

o  i5 

I 

o  i5 

O.OI 

o.i5 

O.60 

o.oo 

2 

3o 

2 

o  3o 

2 

o  3o 

O.O2 

o.3o 

0.61 

9.16 

3 

45 

3 

o  45 

3 

o  45 

o.o3 

o.45 

0.62 

9«3o 

4 

60 

4 

I   0 

4 

I   O 

o.o4 

0.60 

o.63 

9-45 

5 

75 

5 

i  i5 

5 

i  i5 

o.o5 

o.75 

o.64 

9.60 

6 

90 

6 

i  3o 

6 

i  3o 

0.06 

o.9o 

o.65 

9.75 

7 

!05 

7 

i  45 

7 

i  45 

0.07 

.o5 

0.66 

9.90 

8 

120 

8 

2   O 

8 

2   0 

0.08 

.20 

0.67 

io.o5 

9 

i35 

9 

2  15 

9 

2  15 

0.09 

.35 

0.68 

IO.2O 

10 

i5o 

10 

2  30 

10 

2  3o 

O.  IO 

.5o 

0.69 

10.35 

ii 

i65 

ii 

2  45 

ii 

2  45 

O.II 

.65 

0.70 

10.  5o 

12 

180 

12 

3  o 

12 

3  o 

O.  12 

.80 

0.71 

10.65 

i3 

i95 

i3 

3  i5 

i3 

3  i5 

o.i3 

.95 

0.72 

10.80 

i4 

210 

i4 

3  3o 

i4 

3  3o 

o.i4 

2.  IO 

0.73 

10.95 

i5 

225 

i5 

3  45 

i5 

3  45 

o.i5 

2.25 

0.74 

II  .10 

16 

240 

16 

4  o 

16 

4  o 

0.16 

2.40 

0.75 

11.25 

i? 

255 

17 

4  i5 

17 

4  i5 

0.17 

2.55 

0.76 

ii.  4o 

18 

270 

18 

4  3o 

18 

4  3o 

0.18 

2.70 

0.77 

H.55 

19. 

285 

J9 

4  45 

:9 

4  45 

o.  19 

2.85 

0.78 

ii  .70 

20 

3oo 

20 

5  o 

20 

5  o 

0.20 

3.oo 

0.79 

n.85 

21 

3i5 

21 

5  i5 

21 

5  i5 

0.21 

3.i5 

0.80 

1  2  .  OO 

22 

33o 

22 

5  3o 

22 

5  3o 

O.22 

3.3o 

0.81 

12.  l5 

23 

345 

23 

5  45 

23 

5  45 

0.23 

3.45 

0.82 

12.  30 

24 

36o 

24 

6  o 

24 

6  o 

0.24 

3.6o 

.0.83 

12.45 

25 

6  i5 

25 

6  i5 

0.25 

3.75 

o.84 

12.  60 

26 

6  3o 

26 

6  3o 

O.26 

3.90 

o.85 

12.75 

27 

6  45 

27 

6  45 

0.27 

4.o5 

0.86 

12.90 

28 

7  o 

28 

7  o 

0.28 

4.20 

0.87 

i3.o5 

29 

7  i5 

29 

7  i5 

0.29 

4.35 

0.88 

I  3.  20 

3o 

7  3o 

3o 

7  3o 

o.3o 

4.5o 

0.89 

i3.35 

3i 

7  45 

3i 

7  45 

o.3i 

4.65 

0.90 

i3.5o 

32 

8  o 

32 

8  o 

0.32 

4.8o 

0.91 

i3.65 

33 

8  i5 

33 

8  i5 

o.33 

4.95 

0.92 

i3.8o 

34 

8  3o 

34 

8  3o 

o.34 

5.io 

0.93 

i3.95 

35 

8  45 

35 

8  45 

o.35 

5.25 

0.94 

14.10 

36 

9  ° 

36 

9  ° 

o.36 

5.4o 

0.95 

i4.25 

37 

9  i5 

37 

9  i5 

0.37 

5.55 

0.96 

i4.4o 

38 

9  3o 

38 

9  3o 

o.38 

5.70 

0.97 

i4.55 

39 

9  45 

39 

9  45 

o.39 

5.85 

0.98 

14.70 

4o 

10  o 

4o 

IO   O 

o.4o 

6.00 

0.99 

i4.85 

4i 

/ 

10  i5 

4i 

10  i5 

0.41 

6.i5 

I.  00 

i5.oo 

42 

10  3o 

42 

10  3o 

0.42 

6.3o 

43 

10  45 

43 

10  45 

o.43 

6.45 

44 

II   0 

44 

II   0 

o.44 

6.60 

45 

ii  i5 

45 

ii  i5 

o.45 

6.75 

46 

ii  3o 

46 

ii  3o 

0.46 

6.90 

47 

ii  45 

47 

ii  45 

0.47 

7.o5 

48 
£ 

12   O 
12  l5 

48 
^ 

12   O 
12  l5 

0.48 
o.49 

7.20 
7.35 

Thous.  of 
seconds  of 
Time. 

Arc. 

5o 

12  30 

5o 

12  3o 

o.5o 

7.5o 

5i 

12  45 

5i 

12  45 

o.5i 

7.65 

O.OOI 

o.oiS 

52 

i3  o 

52 

i3  o 

O.52 

7.80 

.OO2 

o.o3o 

53 

i3  i5 

53 

i3  i5 

o.53 

7-95 

.oo3 

o.o45 

54 

1  3  3o 

54 

i3  3o 

o.54 

8.10 

.oo4 

0.060 

55 

i3  45 

55 

i3  45 

o.55 

8.25 

.oo5 

0.075 

56 

i4  o 

56 

i4  o 

o.56 

8.40 

.006 

0.090 

57 

i4  1  5 

57 

i4  i5 

0.57 

8.55 

.007 

o.  io5 

58 

i4  3o 

58 

i4  3o 

o.58 

8.70 

.008 

O.  120 

59 

i4  45 

59 

i4  45 

o.59 

8.85 

.009 

o.i35 

• 

60 

i5  o 

60 

i5  o 

0.60 

9.00 

.010 

o.  i5o 

364 


TABLE    YII I. — B ESSEL'S    REFRACTIONS. 


l?tp- 

Mean 
Refract. 

Differ, 
for  10'. 

Log.  A. 

Diff. 

M. 

N. 

llf: 

Mean 
Refract. 

Diff. 

Log.  A. 

M. 

—  1 

N. 

0     i 

/    „ 

» 

0      / 

/   // 

" 

0   0 

34  54.1 

I  .  1069 

1.7344 

10   0 

5  16.2 

5.o 

1.74623 

i  .oo4i 

.0420 

10 

32  49-2 

124.9 

o.758o3 

I  .0962 

I  .6767 

10 

5  II.  2 

A  8 

I  .74670 

i  .oo4o 

.0409 

20 

3o  52.3 

110.9 

T  /-»ft    ft 

.o3248 

I.  0860 

I  .6262 

20 

5  6.4 

/|  *  U 

I.747I4 

i  .oo39 

.0398 

3o 

29  3.5 

I  OO  .  o 
rr»r»  ft 

.18228 

1.0780 

1.6789 

3o 

5  1.7 

4-7 
A  5 

1.74767 

i.oo38 

.o387 

4o 
5o 

27  22.7 
26  49.8 

I  OO  .  o 
92.9 

.28137 
.353oo 

9909 
7i63 

I  .0710 
i.o648 

1.5373 

1.4996 

4o 
5o 

4  67.2 
4  62.8 

4  .  -J 

4.4 
A  3 

1.74799 

i.74839 

i  .oo37 
i.oo36 

.o377 
.o367 

I   0 

24  24.6 

0.2 

.40764 

,o 

1.0693 

1.4653 

II   O 

4  48.5 

i|  •  *J 

A  2 

1.74876 

i.oo35 

.o357 

10 

23   6.7 

77-9 

.45o86 

35  r 

i.o546 

i.434i 

IO 

4  44.3 

£f  V<* 

/    T 

1.74912 

i.oo34 

.o347 

20 

21  55.6 

7i  .1 
p./.  „ 

.48602 

2028 

i  .0606 

1.4067 

20 

4  4o.2 

4.1 

1.74947 

i.oo33 

.o338 

3o 

4o 

20  60.9 
19  61.9 

04.  7 
69.0 

^3  n 

.5i53o 
i  .54oio 

2480 

i.o465 
i  .0429 

i.3797 
i.356o 

3o 

4o 

4  36.3 
4  32.4 

s:9 

3  7 

1.74981 
i  .76013 

I  .O032 

i  .oo3i 

.0828 
,o3i8 

5o 

18  58.o 

00.9 

4n  / 

i.  66142 

i.o397 

1.3342 

5o 

4  28.7 

J.  J 

Q  _ 

1.76043 

i  .oo3o 

.o3o8 

2   0 

18  8.0 

9.4 

A5  6 

1.67996 

1623 

i  .o368 

i  ,3i4i 

12   0 

4  26.0 

.5.7 

3.6 

i  .  76072 

i  .oo3o 

.0299 

10 

17  23.0 

L+-J  •  \J 

1.69618 

i.o342 

1.2966 

10 

4  21.4 

3  A 

i  .76101 

i  .0029 

.0290 

20 

i  6  40.7 

39.8 

i  .6io4i 

1237 

i.o3i8 

1.2783 

20 

4  18.0 

<J  •  4 

3.4 

i  .76129 

i  .0028 

.0281 

3o 

16  0.9 

37.  5 

1.62278 

<ij  / 
[  076 

i  .0298 

i  .2624 

3o 

4  14.6 

3.3 

i.75i55 

I  .O027 

.O272 

4o 

16  23.4 

35  fi 

1.63353 

j.  <->  i  ~j 

i  .0278 

1.2477 

4o 

4  ii.  3 

3.3 

1.76180 

I  .0027 

.0264 

5o 

i4  47-8 

O  J  •  U 

33  o 

1.64286 

828 

i  .0261 

I.234I 

5o 

4  8.0 

3.  i 

i  .76206 

I  .O026 

.0268 

\   0 

i4  i4-6 

O  O  •  -s 

i.65n4 

i  .0244 

i  .2216 

i3  o 

4  4.9 

3.  i 

i  .76229 

I  .OO26 

.0262 

IO 

i3  43.7 

3o  .9 

r,Q   „ 

1.66869 

60  I 

I  .O23o 

i  .2098 

10 

4  1.8 

3.o 

i  .76262 

I  .OO26 

.0246 

20 

i3  16.0 

20.7 

o  (\  n 

1.66660 

GAA 

I  .0216 

i  .  i989 

20 

3  58.8 

1.76274 

I  .OO25 

.0241 

3o 

12  48.3 

20.7 
24.6 

i  .67204 

V^4| 

6OQ 

I  .O2O4 

1.1888 

3o 

3  55.9 

2  .  O 

i  .76296 

I  .0024 

.0235 

4o 

12  23.7 

23.  O 

i.678i3 

57O 

I  .0192 

I.I794 

4o 

3  53.o 

2.8 

i.753i6 

I  .OO24 

.0230 

5o 

12   0.7 

0  T   ft 

1.68383 

^  1  \J 

I  .0182 

i  .  1706 

5o 

3  60.2 

r,   ft 

i.75336 

1.0023 

.0226 

4  o 

ii  38.9 

2  I  .  o 

1.68908 

A   § 

I  .0172 

1.1624 

i4  o 

3  47-4 

2  .  O 

2  7 

1.75355 

.0220 

IO 

ii  i8.3 

20  .  o 

1.69384 

A3 

i.oi63 

i.i549 

10 

3  44-7 

*  •  / 
2.6 

1.75373 

.0216 

20 

10  58.6 

19.7 

1.69816 

379 

i.oi55 

1.1478 

20 

3  42.1 

2.6 

i.7539i 

.0212 

3o 

10  39.6 

19.0 

ift  / 

1.70188 

072 
3t7 

1.0147 

i.i4o8 

3o 

3  39.5 

2.5 

1.76408 

.0208 

4o 

10  21.2 

10.4 

i  .  70606 

017 

i  .oi4o 

i  .  i  342 

4o 

3  37.o 

2.5 

i  .  76426 

.0204 

5o 

10  3.3 

17.9 

rfi  ft 

1.70772 

248 

i.oi33 

i.  1283 

5o 

3  34.5 

2  A 

1.76441 

.02OO 

5  o 

9  46.5 

I  U  .  O 

I  .  7I020 

i  .0127 

I  .  I229 

16  o 

3  32.i 

•  «4r 

i3.5 

1.76467 

.0197 

10 

9  3o.9 

I  D  .  O 
a~ 

1.71279 

243 

i  .0121 

I.II78 

16 

3  18.6 

I  2  .  O 

1.76643 

.0176 

20 

9  16.0 

•  9 
T  /  T 

I  .71622 

oo  rj 

i  .0116 

I  .Il3o 

17 

3  6.6 

10.  8 

1.76616 

.0166 

3o 

9  1.9 

14*1 

I.7I749 

44  / 

212 

I  .OIIO 

I  .  IO82 

18 

2  55.8 

917 

i  .  76676 

.oi39 

4o 

8  48.4 

T  9  8 

I.  7I96l 

T  on 

i  .0106 

i.io36 

19 

2  46.i 

•  / 
8.8 

1.76726 

.0124 

5o 

8  35.6 

1  2  .  O 

0 

i  .-72160 

199 

i  .0100 

i  .0992 

20 

2  37.3 

ft  n 

1.76771 

.0111 

6  o 

8  23.3 

12  .  o 

T  T   n 

1.72346 

173 

i  .0096 

i  .0961 

21 

2  29.3 

o  .  O 

7  A 

i  .76809 

.0101 

10 

8  ii.  6 

ii.  7 

T  T    "^ 

1.72619 

1  /O 

i  .0092 

1.0914 

22 

2  21  .  9 

1  •,** 

fi  7 

1.76842 

.0092 

20 

8  o.3 

I  I  .  0 

Tr.  0 

1.72681 

I  02 
T^T 

i.  0088 

1.0879 

23 

2  l5  .2 

U.  / 

6.3 

1.76871 

.oo83 

3o 

7  49.5 

I  O  •  o 
Tr>  3 

.72832 

1  O  1 

T  /O 

i  .  0084 

i.o846 

24 

2   8.9 

5  7 

1.76897 

.0076 

4o 

7  39.2 

I  O  .  O 

.72974 

142 
TQT 

i  .0081 

1.0816 

25 

2   3.2 

0  .  / 

c  / 

1.76919 

.0068 

5o 

7  29.2 

IO  .  O 
_  c 

.73io5 

I  O  I 

TO/ 

i  .0078 

i  .0784 

26 

57.8 

J  .  4 
5  n 

.oo63 

7  ° 

7  19-7 

9.5 

.73229 

124 

118 

i  .0076 

i  .0764 

27 

52.8 

*J  •  O 

A  6 

1.75967 

.0068 

10 

7  10.6 

9.2 

ft  ft 

.73347 

i  .0078 

i  .0726 

28 

48.2 

*4  •  v 

A  A 

i.75973 

.0064 

20 

7  i-7 

O  .  O 
8/ 

.73459 

112 

i  .0070 

1.0697 

29 

43.8 

4.4 

A  T 

1.76988 

.oo49 

3o 

6  53.3 

.4 

ft   r> 

•73564 

10 

1.0067 

1.0671 

3o 

39.7 

ftf  ••  I 

3  o 

i  .76ooi 

.oo46 

4o 

6  45.i 

O  .  2 

.73663 

99 

i  .0066 

i.o646 

3i 

35.8 

0.9 

3  7 

i  .76oi2 

.oo43 

5o 

6  37.2 

7-9 

.73757 

00 

i  .0062 

i  .0622 

32 

32.1 

*•  / 

3/ 

1.76023 

.oo4o 

8  o 

6  29.0 

7.6 

.73845 

00 
00 

i  .0060 

i  .0600 

33 

28.7 

«4 
3  3 

1.76033 

.oo37 

IO 

6  22.3 

7.3 

.73928 

oO 

i.  0068 

1.0679 

34 

26.4 

o  .  o 

3T 

i  .76042 

.oo34 

20 

6  16.2 

7-  * 
fi  » 

.74007 

79 

nfi 

i  .0066 

i.  0669 

35 

22.3 

•  1 

3  0 

i  .  76060 

.oo3i 

3o 

6  8.4 

u  .  o 
fi  ft 

.74o83 

7° 

i  .0064 

i.o54o 

36 

19.3 

•J  •  U 

2.8 

1.76068 

.OO29 

4o 

6  1.8 

U  «  tJ 

6/ 

.74i55 

72 
fifi 

i  .0062 

1.0623 

37 

i6.5 

o   f-7 

1.76066 

.0027 

5o 

5  55.4 

•  4 

6   T 

.74223 

uo 

i  .0060 

i  .0608 

38 

i3.8 

-  •  / 

2.6 

i  .  76071 

.0026 

9  ° 

5  49-3 

U  •  JL 
fi  0 

.74288 

fi/l 

i  .  0049 

1.0493 

39 

II  .2 

2.5 

1.76077 

.0025 

10 

5  43.3 

U  •  (_ 

5   7 

.74352 

U4 
fin 

i  .0047 

i  .0479 

4o 

8.7 

2  .  / 

i  .76082 

.0023 

20 

5  37.6 

0.7 
K  C 

.74412 

UD 
C£ 

i  .0046 

i.o466 

4i 

6.3 

i  .76087 

.0021 

3o 

5  32.o 

O  .  O 

5  5 

.74468 

OO 

£3 

i.  oo45 

i.o454 

4a 

4.o 

2  2 

i  .  760152 

.0020 

4o 

5  26.6 

c  0 

i  .  74621 

O  0 

i.oo43 

i  .0442 

43 

1.8 

O   T 

i  .76o96 

.0019 

5o 

5  2!.3 

9  •  a 

i.74573 

5o 

i  .0042 

i.o43i 

44 

o  69.7 

2.O 

i  .76ioo 

• 

.0019 

TABLE   VIII. — BESSEL'S    REFRACTIONS. 


365 


App. 
Alt. 

Mean 
Ref. 

Difl 

Log.  A. 

Factor  B,  depending  on  the 
Barometer. 

Factor  T,  depending  on  the  external  Thermometer. 
Fahrenheit. 

45 
46 

4y 

48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57.7 
55.7 
53.8 
5i.9 

50.2 

48.4 
46.7 
45,i 
43.5 
4i.9 
4o.4 
38.9 

2.( 
I  .( 
I  .C 

I  .  •; 

l,€ 

[    ^  r 

i.e 

i.e 

1.6 

T      C 
I      f 

i.4 
.4 
•  4 
.4 
.  3 
.*3 
.3 

.2 

.3 

.2 
.2 
.2 

.  I 
.2 
.1 
.1 
.  I 
.  I 
.  I 

5.3 
5.i 

5.i 

I  .7610^ 

Eug. 
In. 

B. 

Log.  B. 

Fahr. 
Deg. 

T. 

Log.  T. 

Fahr. 
Deg. 

T. 

Log.  T. 

I  .76107 
I  .76111 
I  .76112 
1.76117 

i  .  76ii9 

I  .  76l22 
I  .76l2^ 

i  .-76126 
i  .76128 
i  .76130 
i.76i32 

27.9 

28.0 
28.1 
28.2 
28.3 
28.4 
28.5 
28.6 
28.7 
28.8 

0.943 
0.946 
0.949 
0.953 

o.956 
0.960 
0.963 
0.966 
0.970 
o.973 

—  0.0256^ 

O.O24o9 
O.0225^ 
0.02099 

—  0.01946 
—  o.oi793 
—  o.oi64o 
—o.oi488 
—0.01336 
—  o,ou85 

0 

20 
19 

18 

17 
16 
i5 
i4 
i3 

12 
II 

i.i56 
i.i53 
i.i5o 
i.i48 
i.i45 
i.i43 
i  .  i4o 
i.i38 
i.i35 
i.iSa 

+o.o6279 
+  o.  06181 
+o.o6o83 
+o.o5985 
+o.o5887 
+o.o579o 
+o.o5693 
+o.o5596 
+o.o55oo 
+o.o54o3 

38 

39 
4o 
4i 

42 

43 

44 
45 
46 

47 

I  .022 
I  .OI9 
I  .OI7 

i.oiS 
i.oi3 
i  .on 
i  .oo9 
i  .oo7 
i.ooS 
i  .oo3 

+  0.0092^ 
+  0.00837 
+  O.0075o 
+  0.0o664 
+  0.00578 
+  0.00492 
+o.oo4o6 

+  0.0032O 
+  0.0023/ 

+0.00149 

*7 
58 

«9 

60 

61 
62 
63 
64 
65 
66 
67 
68 

37.5 
36.i 

34.7 
33.3 

32.0 

3o,7 
29.4 
28.2 
26.9 
25.7 
24.5 
23.3 

i  .76134 
1.76136 
i.76i38 
i.76i39 
1.76140 
1.76142 
i.  76i43 
1.76144 
1.76145 
1.76146 
1.76147 
1.76148 

28.  9 

29.0 
29.I 
29.2 
29.3 
29-4 
29.5 
29.6 
29.7 
.9.8 

9-9 
3o.o 

o.976 
0.980 
0.983 
0.987 
0.990 
o.993 

°-997 
i  .000 
i.ooS 
i  .007 
i  .010 
i  .oi4 

—  o.oio35 
—o.oo885 
—  0.00735 
—  o.oo586 
—  o.oo438 

O.OO29O 
0.00142 

+o.oooo5 
+o.ooi5i 

+  0.00297 

+o.  oo443 
+o.oo588 

10 

c 

8 

6 

f 

L 

2 
—    I 
0 

+  I 

i.iSo 

1.128 

I.  125 
I  .123 
I  .  120 

1.118 
i.n5 
i  .n3 

I  .  IIO 

i  .  1  08 
1.106 
i.ioS 

+  o  .o53o7 

+  0.o52I  I 

+o.o5n5 
+o.o5o2o 
+0.04924 
+0.04820. 
+o.o4734 
+o.o464o 
+o.o4545 
+o.o445i 
+  o.o4357 
+  o.o4263 

48 

^ 
5o 

5i 

52 

53 
54 
55 
56 

57 
58 
59 

I  .001 

i  .000 
o.998 
0.996 
0.994 
0.992 
0.990 
0.988 
0.986 
o.984 
0.982 
0.980 

+  0.0006^ 
—  O.OO02I 

—  0.00106 
—  0.00191 
—  0.00275 
—  o.oo36o 
—  o.oo44/ 
—  o.oo528 
—  0.00612 
—  0.00698 
—  0.00780 
—  o.oo863 

69 
70 

7i 
72 

73 
74 
75 
80 
85 
90 

22  .2 
21  .O 
E9.9 

[8.8 
[7.7 
[6.6 
[5.5 

[0.2 

5.i 

0.0 

1.76148 
1.76149 
.  76150 
•76i5o 
•76i5i 
•76i5i 
.76152 
.76154 
i.76i56 
i.76i56 

O.I 
0.2 

o.3 
o.4 
o.5 
0.6 
0.7 
0.8 
0.9 
i  .0 

1.017 

I  .020 
.024 
.027 

.o3i 
.o34 
.o37 
.o4i 
.o44 
.047 

+0.00732 
+0.00876 

+  O.OIO2O 

+o.on63 
+o.oi3o6 
+0.01448 
+o.oi589 
+0.01731 
+0.01871 

+  O.O2OI2 

2 

0 

4 
5 
6 

7 

8 

9 

10 

ii 

12 

i3 

I  .  101 

1.098 
i  .096 
i  .094 
i  .091 
i  .089 
1.087 
1.084 
1.082 
1.080 
1.078 
i  .o75 

+  0.04169 
+o.o4o76 
+  o.o3982 
+o.o3889 
+o.o3796 
+o.o37o4 
+o.o36ii 
+o.o35i9 
+o.o3427 
+o.o3335 
+o.o3243 
+o.o3i52 

60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 

o.978 
0.977 
o.975 
o.973 
°-97i 
0-969 
o.967 
o.965 
o.964 
o.962 
o.96o 
o.958 

—  o.oo946 

O.OI029 
O.OIII2 

—  o.on95 
—  0.01278 
—  o.oi36o 
—  o.oi443 
—  o.oi525 
—  0.01607 
—  o.oi689 
—  0.01770 
—  o.oi852 
—  o.oi933 

O.O2Ol5 

—  0.02096 
—  0.02177 

0.02257 

—0.02338 
—  0.02410 

i4 
i5 
16 

17 

18 

*9 
20 

21 
22 
23 
24 
25 

i.o73 

1.071 
i  .069 
i.  066 
i.o64 
1.062 
i.  060 
r.o57 
i.o55 
i.o53 
i.oSi 
i.o49 

+o.o3o6o 
+o.o2969 
+0.028-78 

+  0.02787 

+0.0269-7 
+0.02606 

+  0.025l4 

+0.02426 
+0.02336 

+  O.02247 
+  O.O2l57 
+  0.02068 

72 
73 

1 
75 

76 

77 
78 

79 
80 

81 
82 
83 

o.956 
o.955 
o.953 
o.95i 
o.949 
o.948 
o.946 
o.944 

0.942 
0.94l 

o.939 
o.937 

Factor  t,  depending  on  the  attached  Thermometer. 
Fahrenheit. 

Fahr. 
Deg. 

t. 

Log.  t. 

Ft. 
Deg. 

t. 

Log.  t. 

0 

—  20 

—  15 

10 

—  5 
o 
+  5 

+  10 

+  i5 

+  20 
+  25 

+  3o 
+35 

.oo5  - 
.oo4  - 
.oo4  - 
.oo3n 
.oo3  - 
.oo3  - 
.002  - 
.002  - 
.001  - 
.001  - 
.000  - 
.000  - 

-O.O0203 

-o.ooi83 
-o.  00164 
-o.ooi44 

-0.00125 

-o.ooioS 
-0.00086 
-0.00066 
-o.  00047 
-0.00027 
-0.00008 

-O  .OOOI  I 

4o 
45 
5o 
55 
60 
65 
70 

75 
80 

85 

9° 
95 

0.999 
0.999 
0.998 
0.998 
0.997 
0.997 
0.997 
0.996 
0.996 
o.995 
o.995 
o.994 

—  o.oooSi 
—  o.ooo5o 
—  o.  00070 
—  o.ooo89 

—  0.02499 
—  0.025-79 
—  0.02609 
—  o.o2738 
—  0.02819 

—  0.00128 
—  o.ooi48 
—  0.00167 
—  0.00186 

—  O.OO2O5 
O.O0225 

—  0.00244 

26 
27 
28 
29 

3o 

3i 

32 

33 
34 
35 
36 

-f-37 

i  .o47 
i.o44 
i  .042 
i  .o4o 
i,o38 
i.o36 
i.o34 

I  .032 

i.o3o 
1.028 
i  .026 
i  .024 

+  0.0!979 

+0.01890 
+o.  01801 
+o.oi7i3 
+0.01624 
+o.oi536 
+o.oi448 
+o.oi36o 

+  O.OI273 

+o.oii85 
+o.oio98 

[-O.OIOII 

84 
85 
86 
87 
88 
89 
90 

91 

92 

93 

9^ 
95 

o.935 
o.934 
o.932 
o.93o 
0.929 

3     927 
3.925 
3.924 
3.922 
3.920 
3.9I9 
3.917 

—  0.02898 
—  o.o2978 
—  o.o3o57 
—  o.o3i36 
—  o.o32i6 
—  0.03294 
~o.o3373 
—  o.o3452 
•—  o.o353o 
—  0.03609 
—  o.o3687 
—  o.o3765 

jog.  Refraction  —log.  cotang.  App.  Alt.+log.  A 
+M(log.  B+log.  0+N  log.  T. 

True  Refraction=Mean  Refraction  X  B  X  t  X  T. 

366       TABLE    IX. — COEFFICIENTS    OF   THE    ERRORS    OF   THE 


N.P.D. 

Azimuth. 

Diff. 

Level.     |  Diff.       Collirn. 

N.P.D.   Azimuth.  |  Diff. 

Level. 

Diff.       Collim. 

Diff. 

i4o 

+  1.555 

n^Q 

+o.o3o 

+  I.556 

80 

+o.49i 

+  0.889 

+  i  .oi5 

i39 

+  I.523 

r>3r 

+  o.o56 

+  1.524 

79 

+0.477 

+  0.900 

+  1.019 

i38 
i37 
i36 
i35 
1  34 
i33 
i3a 

+  1  .492 

+  I.463 
+  I.434 
+  i  .4o6 

+  I.354 

+  1  .  329 

+  i.3o4 

.029 
.029 
.028 
.027 
,025 
.025 
.025 

+  0.081 
+  o.  io5 
+0.128 
+o.  i5o 

+  0.  I72 

+o.i93 

+  0.2I3 
+  0.232 

.024 

.023 
.022 
.022 
.021 
.020 
.019 

+  1.466 
+  i  .44o 
+  i.4i4 
+  i  .390 
+  1.367 
+  1.346 
+  I.325 

78 

77 

74 
73 
72 

+0.462 
+0.448 
+o.434 
+0.419 
+o.4o5 
+o.39o 
+o.375 
+o.36o 

.014 
.014 
.oi5 
.014 
.oi5 
.oi5 
.oi5 

T  r 

+  0.912 
+  0.923 

+o.935 
+0.947 
+0.958 
+0.970 
+0.982 
+o.994 

.Oil 
.012 
.012 
.Oil 
.OI2 
.OI2 
.012 
_  T  q 

+  1  .022 
+  1  .026 

+  i.o3i 
+  i.o35 
+  i  .o4o 
+  1.046 
+  i.o5i 
+  i.o58 

.oo4 
.oo5 
.oo4 
.oo5 
.006 
.oo5 
.007 
c 

i3o 

+  i  .281 

00  ^ 

+  O.  25l 

+  i  .3o5 

7° 

+0.345 

+  i  .007 

I29 

+  I.258 

+  0.270 

or  8 

+  1.287 

69 

+o.  329 

+  1  .OI9 

OT°) 

4-1.071  '5J2J 

128 
I27 
126 
125 
!24 
123 
122 
121 

+  1.236 

+  1  .2l4 

+  i  .  i93 
+  1  .  173 
+  i.i53 
+  i.i33 

+  i  .n4 
+  1.096 

.  O22 
.021 
.020 
.020 
.O2O 
.OI9 
.018 

+  0.288 

+o.3o5 

+  0.322 

+o.339 
+0.355 
+o.37i 
+0.386 
+  o.4oi 

.017 
.017 
.017 
.Ol6 
.Ol6 

.oi5 
.oi5 

_^_  c 

+  i  .269 

+  1  .^52 

+  I.236 

+  J  .221 
+  1  .206 
+  1  .  192 
+  1  .  179 
+  1  .  l67 

68 

67 
66 
65 
64 
63 
62 
61 

+o.3i3 
+0.298 
+0.281 
+0.265 
+0.248 

+  0.23l 
+  0.2l4 

+o.  196 

.oi5 
.oi7 
.016 
.017 
.017 
.oi7 

.018 

+  1.032 

+  1.045 

+  i.o58 
+  1-071 
+  i.o85 
+  i  -o98 

+  1  .  I  12 

+  i  .  126 

.oi3 
.oi3 
.oi3 

.oi4 

.0!3 

.014 
.014 

n  r  c 

+  i  .079 
+  1.086 
+  i  .095 
+  i.io3 
+  i.n3 

+  1  .  122 

+  i.i33 
+  i.i43 

.007 
.009 
.008 

.010 

.009 

.Oil 

.010 

120 

n8 

117 

116 
ii3 

112 
III 

+  1.077 
+  i  .059 
+  i  .042 
+  i  .024 
+  i  .007 
+0.991 
+0.974 
+o.958 

+  0.942 
+  0.927 

.018 
.OI7 
.018 
.017 
.016 
OI7 

.016 
.016 
.oi5 

r\r  fi 

+  o.4i6 
+o.43o 
+o.444 
+o.458 

+  0.472 

+  0.486 
+°«499 

+0.5l2 

+o.525 
+o.537 

.014 
.014 
.oi4 
.014 
.014 
.oi3 
.oi3 
.oi3 

.012 
_  T  q 

+  i.i55 
+  i.i43 
+  i.i33 

+  1  .  122 
+  I.II3 

+  i  .  io3 
+  i.o95 
+  1.086 
+  1.079 
+  1.071 

60 
59 

58 

57 
56 
55 
54 
53 

52 

5i 

+o.i79 
+0.160 

+  0.  l42 
+  0.  122 
+  0.103 

+o.o83 
+  0.062 
+o.o4i 

+  0.020 
0.002 

.019 

.018 

.020 
.019 
.O2O 
.021 
.021 
.021 
.022 

+  i.i56 
+  1.186 

+  1  .202 
+  I.2I8 

+  1.234 

+  1.252 
+  1  .269 

+1.287 

.oi5 
.oi5 

!oi6 

.016 
.016 
.018 
.017 
.018 

„  T  o 

+  i.i55 
+  i  .  167 
+  1.179 
+  1.192 
+  1.206 

+  1  -221 
+  1.236 
+  1  .252 
+  1  .269 
+  1.287 

.012 
.012 
.013 

.oi4 
,oi5 
.oi5 
.016 
.017 
.018 

_  T  o 

I  IO 

+  o  .911 

+o.55o 

+  i  .  064 

5o 

O.O25 

+  i  .3o5 

+  i.3o5 

I09 

I  08 

+  0.896 
+  0.88I 

.oi5 
.oi5 
oi5 

+0.562 
+o.574 

.012 
.OI2 

+  i.o58 
+  i  .o5i 

49 

48 

—  0.049 
—  0.073 

.024 
.024 

+  I.324 
-j-i.344 

.019 
.020 

+  I.325 
+  1.346 

.020 

.021 

I07 

+0.866 

oi5 

+0.586 

Ol3 

+  i  .o46 

47 

—  0.098 

0^6 

-j-i.364 

+  i  .367 

.021 

I  06 

io5 

+o.85i 
+o.836 

.oi5 

+o.599 
+0.610 

.Oil 

+  i  .o4o 
+  i.o35 

46 
45 

0.  124 

—  o.  i5o 

.026 

+  I.385 
+  i.4o6 

.O2I 

+  i  .  Sgo 

-fi.4x4 

.  O24 

io4 
io3 

102 
101 

+0.822 
+0.808 
+0.793 
+0.779 

.014 
.014 
.oi5 

.014 
_  T  / 

+o.  622 
+0.633 
+0.645 
+0.656 

.  OI2 
.Oil 
.OI2 
.Oil 

+  i.o3i 
+  1.026 

+  1  .022 
+  1  .019 

44 
43 

42 

4i 

—  0.178 

O.  2O7 
—  0.237 
0.267 

.  O2O 

.029 

.o3o 
.o3o 

_.  qq 

+  1.428 
+  i  .452 
+  1.476 
+  i  .5oi 

.022 
.024 
.024 
.025 

+  i  .44o 

+  1.466 
+  i  .494 
+  i  .524 

.026 

.028 

.o3o 

100 

99 

98 

97 
96 

+0.765 
+o.75i 

+o.737 
+0.723 
+0.710 

.oi4 
.014 
.014 
.oi3 

+0.668 
+0.670. 
+o.  69o 
+0.701 
+0.712 

.oii 

.Oil 
.Oil 
.Oil 

+  i  .oi5 
+  1  .012 
+  1  .010 
+  1.008 
+  i  .006 

4o 
39 
38 

37 
36 

—  o.3oo 
—0.333 
—0.368 
—  o.4o5 
—0.443 

.o33 
.o35 
.o37 
.o38 

+  i  .527 
+  I.554 
-j-i.582 
+  1.612 
+  I.643 

.027 

.028 

.o3o 
.o3i 
o32 

+  I.556 
+  i.589 
+  1.624 
+  1.662 
+  1.701 

.o33 
.o35 
.o38 
.039 

95 

94 

+0.696 

+0.682 

.014 

OT  ^ 

+o.723 
+o.734 

.Oil 

+  1  .004 
+  1  .002 

35 

34 

—0.484 
—  0.526 

.042 

+  1.675 
+  1.709 

.o34 

+  i.743 
+  1.788 

.o45 

n/Q 

93 

92 

91 

+0.669 
+0.655 
+o.64i 

.014 
.014 

+o.745 
+o.756 
+0.767 

.Oil 
.Oil 

+  1  .OOI 
+  1  .OOI 
+  1  .000 

33 

32 
3! 

—  o.57i 
—  0.618 
—  o.667 

-o47 
.049 

+  I-745 
+  1.783 
+  1.823 

.o38 
.o4o 
rt/o 

+  I.836 
+  1.887 
+  i  .942 

.o5i 
.o55 

.-.CO 

9° 

89 

88 
87 
86 
85 
84 
83 
82 
81 

+0.628 
+0.614 
+0.601 
+o.587 
+o.573 
+o.56o 
+0.546 
+0.532 
+o.5i9 
+o.5o5 

.  OIO 

.014 
.oi3 
.oi4 
.014 
.oi3 
.014 
.014 
.oi3 
.oi4 

+0.778 
+o.789 
+0.800 
+0.811 
+0.822 
+0.833 
+o.844 
+0.855 
+0.867 
+0.878 

.01  1 

.Oil 
.Oil 
.Oil 
.Oil 
.Oil 
.Oil 
.Oil 
.012 
.Oil 

+  1  .000 

+  1  .000 
+  1  .001 
+  1  .001 

+  1  .002 

+  I  .  004 

+  1.006 
+  1.008 

+  1  .010 
+  1  .012 

3o 

29 

28 

27 
26 
25 
24 
23 
22 
21 

0.720 

—  o.776 
—0.836 
—  0.900 
—0.968 

—  I  .  120 
1  .206 
—  1.298 
—  I  .400 

.o56 
.060 
.064 
.068 
.o73 

•°79 
.086 
.092 

.  102 

+  1,866 
+  i  .9ii 
+  1.959 

+  2.  Oil 
+  2.066 
+  2.125 
+  2.l89 
+  2.257 
+  2.332 

+2.4i4 

.045 
.048 

.052 

.o55 
.o59 

.064 
.068 
.075 
.082 

+  2  .OOO 

+  2.o63 
+2.i3o 

+  2.2O3 

+2.281 

+2.366 
+2.459 
+  2.559 

+2.669 

+2.790 

.o63 
.067 
.073 
.078 
.o85 
.o93 

.  IOO 
.  IIO 
.  121 

TRANSIT   INSTRUMENT  FOR  WASHINGTON   OBSERVATORY.  367 


Polaris.     Upper  Culmination. 

N.P.D. 

Azimuth. 

Diff. 

Level,       i  Diff. 

Collimation.    Diff. 

N.P.  Dist.     Azimuth.          Level.         Collimation. 

4-21 

20 

—   i.4oo 
-   i.5n 

.Ill 

4  2.414 
4-  2.5o3 

.089 

—  |—    2  •  7QO 

4-  2.924 

.i34 

T   /A 

I    3l       0 
3o  5o 

—  28.768 
—         .822 

424.493 

4      -536 

437.782 
4      -85i 

1  9 

18 

—   1.633 
—   1.767 

.134 
T  c  r 

4-  2.602 
4  2.711 

.099 

.  109 

4  3.o72 
4   3.236 

.164 

T  o  / 

3o  4o 
3o  3o 

—        .876 

4      .58o 

4     .624 

4     .921 
4      -991 

16 

—   1.918 
—  2.086 

.168 

4-    2.832 

4  2.968 

.i36 

c  / 

4   3.420 

4  3.628 

.208 

3o  20 
3o  10 

29.039 

4-      .668 
4      -712 

4-38.  061 
4      .i3i 

i5 

—  2.277 

.191 

4-    3.122 

.  104 

r 

4   3.864 

.236 

3o     o 

—        .094 

4      -756 

4-      .202 

i4 
i3 

12 

—  2.494 
-  2.743 
—  3.o34 

.217 
.249 
.291 

q  /_ 

4  3.297 
4-  3.498 
4-  3.732 

.175 

.201 
.234 

4  4.i34 
4  4.445 
4  4.810 

.270 
.3n 
.365 

/q 

29  5o 
29  4o 
29  3o 

—      .i5o 

—        .205 

—     .261 

4-     .801 
4     .845 
4-     .890 

4     -272 
4-     -344 
4-     -4x5 

II 
10 

9 

8 

—  3.376 
—  3.786 
—  4.286 
—  4.910 

.  J42 

.4io 
.5oo 

.624 

0 

4-  4.008 
4-  4.34o 
4  4-743 
4-  5.246 

,  270 
.332 
.4o3 
.5o3 

4-  5.241 
4-  5.759 
4  6.392 

4  7-i85 

.431 

.5i8 
.633 
.793 

Pol 

i   3i     o 
3o  5o 

aris.     Lo\ 
43o.o23 

4     -077 

vei  Culmir 

—  22  .936 
—      .980 

iation. 

-37.782 
—      .85i 

7 

-  5.7ii 

.  OOI 

4-  5.892 

.  046 

4  8.206 

3o  A.O 

4         .132 

—  23.023 

—      .921 

6 

—  6.777 

4-  6.752 

.000 

4  9.567 

3o  3o 

4-    .186 

—     .067 

—     .991 

5 

—  8.268 

4-  7-955 

4n.474 

3o  20 

4     .241 

—      .in 

—  38.o6i 

4 

—  10.  502 

4-  9-757 

4i4.336 

3o   10 

-f     .295 

—      .i55 

—      .i3i 

3 

—  14.223 

412.759 

419.107 

3o     o 

4-      .35o 

—     .200 

—        .202 

2 

—  21.660 

+  i8.759 

428.654 

29  5o 

4     .4o5 

—     .244 

—     .272 

4-     I 
0 

—43.961 

436.  75o 

4-57.299 

29  4° 
29  3o 

4=    .461 
4      .5i6 

—     .289 
—      .334 

—     .344 
—     .4i5 

I 

2 

4-45.2I7 
4-22.916 

—  35.i93 

—  17  .202 

—57.299 
—28.654 

/I  Ursae  Minoris.     Upper  Culmination. 

q 
~    -J 

II  .202 

—  19.  107 

i     9     o 

—38.i44 

432.  o57 

449.826 

—  4 

4-n.758 

—     8.201 

—  i4-336 

8  5o 

—     .238 

4-        .132 

4     -946 

—  5 

4  9.524 

—  6.398 

—  ii  .474 

8  4o 

—      .332 

4      .209 

45o.o68 

—  6 

4  8.o33 

—  5.196 

—  9.567 

8  3o 

—      .426 

4      .285 

4     .189 

—   7 

4  6.967 

—  4.336 

.860 

—  8.206 

8    20 

—        .522 

4-      .362 

4        .312 

—  8 

4   6.166 

.8oi 

F\r>  f 

—  3.689 

•  647 

r     o 

-  7.i85 

8   10 

—        .6l8 

4-     -439 

4-     .435 

—  9 

10 

—  ii 

—  13 

—  14 
—  15 

4-  5.042 
4  5.o4i 
4  4.632 

4  4.290 
4  4.000 
4  3.749 
4  3.532 

.5oi 
.409 
.342 
.290 

.25l 

.217 

—   3.i86 
—  2.783 
—  •  2.452 

2  .  I76 

-  .942 

—     .  74o 
—     .565 

.  OOO 

.4o3 
.33i 

.276 
.234 
.202 

.i75 

—  6.392 
-  5.759 
—  5.24i 
—  4.810 
—  4-445 
-  4.i34 
—  3.864 

!e33 

.5i8 
.43i 
.365 
.3n 
.270 

A.  Ursae  Minoris. 

8  5o  -j-      .493 
8  4o4-      .587 
8  3o-h      .682 
8  20  -f      .778 

Lower  C\ 

—  3o.5oo 
—      .576 
—     .652 
—      .729 
—     .806 

ilmination. 
—49.826 
-      -946 
—  5o.o68 
—      .189 

—        .312 

—  16 

4  3.342 

.  190 

T£Q 

-     .4n 

.i54 

—  3.628 

.236 

8   10 

4    -874 

—     .883 

-      .435 

—  17 
—  18 

4-  3.i74 

4-   3.023 

Tqc 

—        .275 

-     .i54 

.  121 

—  3.420 
—  3.236 

.184 

-  C  / 

51  Cephei.     Upper  Culmination. 

—  19 

—  20 
—  21 
—  22 

4-  2.766 
4  2.655 
4  2.554 

.  122 
.III 
.  IOI 

—  i.o45 
—  o.947 
—  o.857 
—  o.776 

.098 
.090 

.O8l 

r 

—  3.072 
—  2.924 
—  2.790 
—  2.669 

.  104 

.i48 
.i34 

.  121 

2    45        0 

44  5o 

44  4o 
44  3o 

—  10  .070 

-  .591 

-    .608 
—    .625 

-f-I^.OOO 

4-      .864 
4     -877 
4     .890 

-(-20.  04,3 

4      .864 
4     .885 
-j-     .906 

—  23 

—24 

—  25 

4-  2.461 

4  2.376 

4    2.297 

,o85 
.079 

—  o.7oi 
—  0.632 

—  0.568 

.O70 

.069 

.064 

—  2.559 
-  2.459 
—  2.366- 

.IIn 
.  IOO 
.  OQ  3 

51  C 

2  45     o 

sphei.     L( 
4-iG.  83i 

)wer  Culrr 

—  12  .293 

ination. 
—20.843 

—26 

—27 

—28 

4-  2.224 

4  a.i55 

-f-     2.092 

.07^ 
.069 
.o63 
/? 

—  o.Sog 

-  0.454 
—  o.4o3 

'o55~  2'281 

r      —  •     2.203 
.  OO  I                      0 
2.  I  JO 

or 

.o85 

.078 
.073 

44  5o 
44  4o 
44  3o 

4     .848 
4-      .864 
-j-     .880 

—      .3o7 

—         .320 

—      .333 

—      .864 
—      .885 
—      .906 

—29 

—  3o 
—  3i 

—  32 

—33 

4"     2.0J2 

4-     .976 
4-     .923 
4-     -873 
4    .826 

.  ooo 
.o56 
.o53 
.o5o 
.047 

-  0.354 
—  0.309 
—  0.26-7 

—    0.227 
—    0.180 

0  0  0  0  0 

(30  O  to  cnvjO 

—  2.o63 

—     2.000 

—     .942 

—     .887 

—      .836 

.067 
.o63 
.o58 
.o55 
.o5i 

6  Ursae 

3     24    20 
24     10 
24       0 

Minoris. 

-12.45: 
-      .46i 

Upper  Cu 

4-H.329 

4-      .338 
4      -347 

[mination. 

4i6.834 

4      .848 
4     .862 

-34 
—35 

4-     .782 
4     •  739 

.o44 
.o43 

—  o.i53 
—  o  118 

.036 
.o35 

—      .788 

.o48 
.o45 

6  Ursae  Minoris.     Lower  Culmination. 

-36 

4    .699 

.o4o 

—  0.086 

.  o'3a 

—      .701 

.042 

3    24    20 

4i3.7o7 

—  9.773 

—  i6.834 

-37 

-38 

4     .661 
4     .624 

.Ooo 

.o37 

—  o.o55 

—    O.O25 

.o3i 
.o3o 

—      .662 
—      .624 

.009 

.o38 

24    10 
24       0 

4    -718 
4     -729 

-      .781 
—      .790 

—     .848 
—     .862 

368  TABLE    X. — REDUCTION   TO   THE   MERIDIAN.     PART  I. 


Sec. 

Om 

1m 

2m. 

3m. 

4m. 

5  m. 

6m. 

7m. 

8m. 

9m. 

10m. 

llm. 

o 

0.0 

2.0 

7.8 

17.7 

3i.4 

49.1 

70.7 

96.2 

125.7 

iSg.  o 

196.3 

237.5 

i 

o.o 

2.0 

8.0 

17.9 

3i.7 

49-4 

71.1 

96.7 

126.2 

i59.6 

197.0 

238.3 

2 

o.o 

2.1 

8.1 

!8.I 

3i.9 

49.7 

7i.5 

97.1 

126.7 

160.2 

197.6 

239.o 

3 

o.o 

2.2 

8.2 

i8.3 

32.2 

5o.i 

71.9 

97.6 

127.2 

160.8 

i98.3 

239.7 

4 

0.0 

2.2 

8.4 

i8.5 

32.5 

5o.4 

72.3 

98.0 

127.8 

161.4 

i98.9 

240.4 

5 

0.0 

2.3 

8.5 

18.7 

32.7 

50.7 

72.7 

98.5 

128.3 

162.0 

i99.6 

24l  .1 

6 

0.0 

2.4 

8.7 

18.9 

33.o 

5i.i 

73.i 

99.0 

128.8 

162.6 

200.  3 

24l.9 

7 

0.0 

2.4 

8.8 

19.1 

33.3 

5i.4 

73.5 

99.4 

129.3 

i63.2 

200.  9 

242.6 

8 

0.0 

2.5 

8.9 

19.3 

33.5 

5i.7 

73.9 

99.9 

129.9 

i63.8 

201  .6 

243.3 

9 

0.0 

2.6 

9.1 

i9.5 

33.8 

52.1 

74.3 

100.4 

i3o.4 

164.4 

2O2  .  2 

244.1 

10 

0. 

2.7 

9.2 

19-7 

34.i 

52.4 

74-7 

100.8 

1  30.9 

i65.o 

202  .9 

244  •  8 

ii 

0. 

2.7 

9.4 

19.9 

34.4 

52.7 

75.i 

101.3 

i3i.5 

i65.6 

2o3.6 

245.5 

12 

0. 

2.8 

9.5 

20.  i 

34.6 

53.i 

75.5 

101  .8 

132.0 

166.2 

204.2 

246.3 

i3 

0. 

2.9 

9.6 

20.3 

34-9 

53.4 

75.9 

102.3 

i32.6 

166.8 

2O4«9 

247.0 

i4 

0. 

3.o 

9.8 

20.  5 

35.2 

53.8 

76.3 

102.7 

i33.i 

167.4 

2o5  .6 

247-7 

i5 

0. 

3.i 

9.9 

20.7 

35.5 

54.i 

76.7 

103.2 

i33.6 

168.0 

206.3 

248.5 

16 

o. 

3.i 

10.  I 

20.9 

35.7 

54.5 

77.1 

103.7 

i34.2 

168.6 

206.  9 

249.2 

17 

O.2 

3.2 

10.2 

21  .2 

36.o 

54.8 

77-5 

104.2 

i34.7 

169.2 

207.6 

249.9 

18 

O.2 

3.3 

IO.4 

21  .4 

36.3 

55.i 

77-9 

104.6 

i35.3 

169.8 

208.3 

250.7 

*9 

0.2 

3.4 

io.5 

21  .6 

36.6 

55.5 

78.3 

io5  .  i 

i35.8 

170.4 

208.  9 

25i.4 

20 

0.2 

3.5 

10.7 

21.8 

36.9 

55.8 

78.8 

io5.6 

i36.3 

171.0 

209.6 

2t>2.2 

21 

0.2 

3.6 

10.8 

22.  O 

37.2 

56.2 

79.2 

106.1 

i36.9 

171.6 

210.3 

253.o 

22 

o.3 

3.7 

II.  0 

22.3 

37-4 

56.5 

79.6 

106.6 

i37.4 

172.2 

211  .O 

253.6 

23 

o.3 

3.8 

I  I  .2 

22.5 

37.7 

56.9 

80.0 

107.0 

i38.o 

172.9 

2II.7 

254.4 

24 

o.3 

3.8 

ii.  3 

22.7 

38.o 

57.3 

80.4 

107.5 

i38.5 

i73.5 

212.3 

255.1 

25 

o.3 

3.9 

ii.  5 

22.9 

38.3 

57.6 

80.8 

108.0 

iSg.  i 

174.1 

213.0 

255.9 

26 

0.4 

4.o 

n.  6 

23.1 

38.6 

58.o 

8i.3 

io8.5 

i39.6 

174.7 

2l3.7 

256.6 

27 

0.4 

4.i 

ii.  8 

23.4 

38.9 

58.3 

SL  7 

109.0 

140.2 

175.3 

214.4 

257.4 

28 

0.4 

4.2 

11.9 

23.6 

39.2 

58.7 

82.1 

109.5 

140.7 

175.9 

2I5.I 

258.1 

29 

o.5 

4.3 

12.  I 

23.8 

39.5 

Sg.o 

82.5 

I  10.  0 

i4i.3 

176.6 

2i5.8 

258.  9 

3o 

o.5 

4.4 

12.3 

24.0 

39.8 

59.4 

83.o 

1  10.4 

i4i.8 

177.2 

216.4 

269.  6 

3i 

o.5 

4.5 

12.4 

24.3 

4o.  i 

59.8 

83.4 

110.9 

142.4 

177.8 

217.1 

260.4 

32 

0.6 

4.6 

12.6 

24.5 

4o.3 

60.  i 

83.8 

in.  4 

i43.o 

178.4 

217.8 

261.1 

33 

0.6 

4-7 

12.8 

24.7 

4o.6 

6o.5 

84.2 

111.9 

i43.5 

179.0 

218.  5 

261.9 

34 

0.6 

4.8 

12.9 

25.0 

40.9 

60.8 

84-7 

112.  4 

i44.i 

179.7 

219.2 

262.6 

35 

0.7 

4.9 

i3.i 

25.2 

41.2 

61  .2 

85.i 

112.9 

i44.6 

i8o.3 

219.9 

263.4 

36 

0.7 

5.o 

i3.3 

25.4 

4i.5 

61.6 

85.5 

n3.4 

i45.2 

180.9 

220.  6 

264.  i 

37 

0.7 

5.i 

i3.4 

25.7 

4i.8 

61.9 

86.0 

113.9 

i45.8 

181.6 

221.3 

264.9 

38 

0.8 

5.2 

i3.6 

25.9 

42.  i 

62.3 

86.4 

n4.4 

i46.3 

182.2 

222.  O 

265.7 

39 

0.8 

5.3 

i3.8 

26.2 

42.5 

62.7 

86.8 

114.9 

146.9 

182.8 

222.7 

266.4 

4o 

0.9 

5.4 

i4.o 

26.4 

42.8 

63.o 

87.3 

n5.4 

147-5 

i»3.5 

223.4 

267.2 

4i 

0.9 

5.6 

i4-i 

26.6 

43.i 

63.4 

87.7 

115.9 

i48.o 

i84.i 

224.1 

268.0 

42 

i  .0 

5.7 

i4.3 

26.9 

43.4 

63.8 

88.1 

116.4 

148.6 

184.7 

224.8 

268.7 

43 

.0 

5.8 

i4.5 

27.1 

43.7 

64.2 

88.6 

116.9 

149.2 

i85.4 

225.5 

269.5 

44 

.  i 

5.9 

i4.7 

27.4 

44.o 

64.5 

89.0 

117.4 

149.7 

186.0 

226.2 

270.3 

45 

.  i 

6.0 

i4.8 

27.6 

44.3 

64-9 

89.5 

117.9 

i5o.3 

186.6 

226.9 

271  .0 

46 

.2 

6.1 

!5.0 

27.9 

44.6 

65.3 

89.9 

118.4 

iSo.g 

187.3 

227.6 

271.8 

4? 

.2 

6.2 

l5.2 

28.1 

44-9 

65.7 

90.3 

118.9 

i5i.5 

187.9 

228.3 

272.6 

48 

.3 

6.4 

i5.4 

28.3 

45.2 

66.0 

90.8 

119.5 

l52.O 

i88.5 

229.0 

273.3 

49 

.3 

6.5 

i5.6 

28.6 

45.5 

66.4 

9I  .2 

120.0 

i52.6 

189.2 

229.  ? 

274.1 

bo 

.4 

6.6 

i5.8 

28.8 

45.9 

66.8 

91.7 

120.5 

i53.2 

189.8 

2^0.4 

274.9 

5i 

.4 

6.7 

i5.9 

29.1 

46.2 

67.2 

92.1 

121  .O 

i53.8 

190.5 

23l.I 

275.6 

52 

.5 

6.8 

16.1 

29.4 

46.5 

67.6 

92.6 

121  .5 

i54.4 

191.1 

23i.8 

276.4 

53 

.5 

7.0 

i6.3 

29.6 

46.8 

68.0 

93.o 

122.  O 

i54.9 

191.8 

232.5 

277.2 

54 

.6 

7-1 

i6.5 

29.9 

47.i 

68.3 

93.5 

122.5 

i55.5 

192.4 

233.2 

278.0 

55 

.6 

7.2 

16.7 

3o.i 

47-5 

68.7 

93.9 

123.  I 

i56.i 

193.1 

234.0 

278.8 

56 

•7 

7.3 

16.9 

3o.4 

47-8 

69.  i 

94-4 

123.6 

i56.7 

193.7 

234.7 

279.5 

5? 

.8 

7.5 

17.1 

3o.6 

48.i 

69.5 

94.8 

124.1 

157.3 

194.4 

235.4 

280.3 

58 

.8 

7.6 

i7.3 

30.9 

48.4 

69.9 

95.3 

124.6 

i57.8 

195.0 

236.i 

281.1 

59 

•9 

7-7 

i7.5 

3l.2 

48.8 

70.3 

95.7 

125.  I 

i58.4 

i95.7 

236.8 

281.9 

TABLE    X. — REDUCTION  TO  THE  MERIDIAN.     PART   I.  369 


Sec. 

12  m.    13  m 

14m. 

15  m, 

16m. 

17m. 

18m. 

19  m.    2O  m.    21  m. 

0 

282.7 

33i.7 

384-7 

44i.6 

5o2.5 

567.2 

635.  9 

7o8.4 

784.9 

865.3 

i 

283.5 

332.6 

385.6 

442.6 

5o3.5 

568.3 

637.o 

•709.4 

786.2 

866.6 

2 

284.3 

333.4 

386.6 

443.6 

5o4.6 

569.4 

638.2 

7io.9 

787.5 

868.0 

3 

285.0 

334.3 

387.5 

444.6 

5o5.6 

57o.5 

639.4 

7I2.I 

788.8 

869.4 

4 

285.8 

335.2 

388.4 

445.6 

506.7 

57i.6 

64o.6 

7i3.4 

790.1 

87o.8 

5 

286.6 

336.0 

389.3 

446.5 

507.7 

572.8 

64i.7 

714.6 

791.4 

872.1 

6 

287.4 

336.  9 

390.2 

447.5 

5o8.8 

573.9 

642.9 

7i5-9 

70.2.7 

873.5 

7 

288.2 

337.7 

391.2 

448.5 

5o9.8 

575.o 

644.1 

7i7.i 

794.o 

874-9 

8 

289.0 

338.6 

392.i 

449.5 

510.9 

576.i 

645.3 

7i8.4 

795.4 

876.3 

9 

289.8 

339.4 

SgS.o 

45o.5 

5i  i  .9 

577.2 

646.5 

719.6 

796.7 

877.6 

10 

290.6 

34o.3 

393.9 

45i.5 

5i3.o 

578.4 

647-7 

720.  9 

79«.o 

fe79.o 

ir 

291.4 

34i.2 

394.9 

452.5 

5i4.o 

579.5 

648.  9 

722.1 

799-3 

880.4 

12 

292.2 

342.0 

395.8 

453.5 

5i5.i 

58o.6 

65o.o 

723.4 

800.  7 

881.8 

i3 

293.0 

342.9 

396.7 

454.5 

5i6.i 

58i.7 

65i.2 

724.6 

802.0 

883.2 

i4 

293.8 

343.7 

397.6 

455.5 

517.2 

582.  9 

652.4 

725.9 

8o3.3 

884.6 

i5 

294.6 

344.6 

398.6 

456.5 

5i8.3 

584.0 

653.6 

727.2 

8o4.6 

886.0 

16 

295.4 

345.5 

399.5 

457.5 

5i9.3 

585.1 

654.8 

•728.4 

806.0 

887.4 

!? 

296.2 

346.4 

4oo.5 

458.5 

520.4 

586.2 

656.0 

729-7 

8o7.3 

888.8 

18 

297.0 

347.2 

4oi.4 

459.5 

521.5 

587.4 

657.2 

73o.9 

808.6 

890.2 

J9 

297.8 

348.1 

402.3 

46o.5 

522.5 

588.5 

658.4 

732.2 

8o9.9 

891.6 

20 

298.6 

349.0 

4o3.3 

46i.5 

523.6 

589.6 

659.6 

733.5 

fen.  3  !  bgS.o 

21 

299.4 

349.8 

404.2 

462.5 

524.6 

590.8 

660.8 

734.7 

812.6 

894.4 

22 

30O.2 

35o.7 

4o5.i 

463.5 

525.  7 

691.9 

662.0 

736.o 

8i3.9 

895.8 

23 

3oi  .0 

35i.6 

4o6.  i 

464.5 

526.8 

593.o 

663.2 

737.3 

8i5.2 

89-7.2 

24 

3oi.8 

352.5 

407.0 

465.5 

527.9 

594.2 

664.4 

738.5 

816.6 

898.6 

25 

3o2.6 

353.3 

4o8.o 

466.5 

528.9 

595.3 

665.6 

739.8 

8i7.9 

900.0 

26 

3o3.5 

354.2 

4o8.9 

467.5 

53o.o 

596.5 

666.8 

74i.i 

8l9.2 

901.4 

27 

3o4.3 

355.i 

4o9.9 

468.5 

53i.i 

597.6 

668.0 

742.3 

820.5 

902.8 

28 

3o5.i 

356.o 

4io.8 

469.5 

532.2 

598.7 

669.2 

743.6 

821.  9 

904.2 

29 

3o5.9 

356.  9 

411.7 

47o.5 

533.2 

599.9 

670.4 

744.9 

823.2 

9o5.6 

3o 

806.7 

^7-7 

412.7 

47i.5 

534.3 

601  .0 

671.6 

746.2 

824.6 

90-7.0 

3i 

3o7.5 

358.6 

4i3.6 

472.6 

535.4 

602.2 

672.8 

747.4 

825.  9 

908.4 

32 

3o8.4 

359.5 

4i4.6 

473.6 

536.5 

6o3.3 

674.1 

748.  7 

827.3 

909.8 

33 

309.2 

36o.4 

4i5.5 

474.6 

537.6 

6o4.5 

675.3 

75o.o 

828.6 

911.2 

34 

3io.o 

36i.3 

4i6.5 

475.6 

538.7 

6o5.6 

676.'5 

75i.3 

829.9 

912.6 

35 

3io.8 

362.2 

4i7.4 

476.6 

539.7 

606.8 

677.7 

752.6 

83i.2 

914.0 

36 

3u.6 

363.1 

4:8.4 

477-  6 

54o.8 

607.  9 

678.9 

753.8 

832.6 

9i5.5 

3? 

3i2.5 

364.0 

4i9.4 

478.7 

54i.9 

6o9.  i 

680.  i 

755.i 

833.  9 

916.9 

38 

3i3.3 

364.8 

420.3 

479-7 

543.0 

610.2 

68i.3 

756.4 

835.3 

oi8.3 

39 

3i4.i 

365.7 

42i.3 

48o.7 

544.i 

611.4 

682.6 

757.7 

836.6    919.  7 

4o 

3i5.o 

366.6 

422.2 

48i.7 

545.2 

6i2.5 

683.8 

769.0 

b^s.o 

921  .  i 

4i 

3i5.8 

367.5 

423.2 

482.8 

546.3 

6i3.7 

685.0 

76o.2 

839.3 

922.5 

42 

3i6.6 

368.4 

424.2 

483.8 

547.4 

6i4.8 

686.2 

76i.5 

84o.7 

923.9 

43 

317-4 

369.3 

425.i 

484-8 

548.4 

616.0 

687.4 

762.8 

842.0 

925.3 

44 

3x8.3 

370.2 

426.  i 

485.8 

549.5 

617.2 

688.  7 

764.i 

843.4 

926.8 

45 

Sig.i 

37i.i 

427.0 

486.  9 

55o.6 

6i8.3 

689.9 

765.4 

844.7 

928.2 

46 

3i9.9 

372.0 

428.0 

487.  9 

55i.7 

619.5 

^691.1 

766.7 

846.1 

929.6 

4? 

320.8 

372.9 

429.O 

488.  9 

552.8 

620.6 

'692.4 

768.o 

847.5 

93i.o 

48 

321.6 

373.8 

429.9 

49o.o 

553.9 

621.8 

693.6 

769.3 

848.9 

932.4 

49 

322.4 

374-7 

43o.9    49i.o 

555.o 

623.0 

694.8 

770.6 

85o.2 

933.8 

5o 

323.3 

375.6 

43i.9 

492.0 

556.1 

624.1 

696.0 

771.9 

85i.6 

935.2 

5i 

324.1 

376.5 

432.8 

493.1 

557.2 

625.3 

697  .3 

773.i 

852.9 

936.6 

52 

325.o 

377.4 

433.8 

494.1 

558.3 

626.5 

698.5 

774.5 

854.3 

938.1 

53 

325.8 

378.3 

434.8 

495.2 

559.4 

627.6 

699.7 

775.8 

855.  7 

939.5 

54 

326.7 

379.3 

435.8 

496.2 

56o.5 

628.8 

701  .0 

777.1 

857.i 

940.9 

55 

327.5 

38o.2 

436.7 

497-2 

56i.6 

63o.o 

702.2 

778.4 

858.4 

942.3 

56 

328.4 

38i.i 

437.7 

498.3 

562.  7 

63i.2 

7o3.5 

779-7 

859.8 

943.8 

5? 

329.2 

382.0 

438.  7 

499-3 

563.  9 

632.3 

704.7 

781.0 

861.1 

945.2 

58 

33o.o 

382.  9 

439.7 

5oo.3 

565.o 

633.5 

7o5.9 

782.3 

862.5 

946.6 

5c>       ^3o.9 

383.8 

44o.6 

5oi.4 

566.1 

634-7 

707.1 

783.6 

863.  9 

948.1 

AA 


370  TABLE    X. — REDUCTION   TO   THE   MERIDIAN.     PART   I. 


Sec. 

22m.  |  23m. 

24m. 

25m. 

26m. 

27  m. 

28m. 

29m. 

30  m.  |  31  m. 

0 

949.6 

io37.8 

1129.9 

1226.  9 

i325.9 

l429.7 

i537.5 

i649.o 

1764.6 

1884.  o 

i 

95i  .0 

io39.3 

n3i.4 

1227.6 

i327.6 

i43i.4 

i539.3 

i65o.9 

1766.6 

1886.  o 

2 

952.4 

i  o4o  .  8 

n33.o 

I229.2 

i329.3 

i433.2 

1641.1 

1662.8 

1768.6 

1888.0 

3 

953.8 

1042.3 

n34.6 

1230.8 

i33i.o 

1434.9 

1642.9 

i654.7 

1770.6 

1890.0 

4 

955.3 

io43.8 

ii36.2 

1232.5 

i332.7 

!436.7 

i544.8 

1666.6 

1892.1 

5 

956.7 

io45.3 

1187.8 

1234.  I 

1334.4 

i438.5 

i546.6 

i658.5 

1774.4 

1894.1 

6 

958.2 

io46.8 

n39.3 

1235.7 

i336.i 

i44o.3 

i548.4 

1660.4 

1776.3 

1896.1 

7 

959.6 

io48.3 

1140.9 

i237.3 

i337.8 

1442.  i 

1660.2 

1662.3 

1778.3 

1898.1 

8 

961  .  i 

io49.8 

n42.5 

I239.O 

i339.5 

i443.9 

1662.1 

l664.  2 

1780.3 

1900.2 

9 

962.6 

io5i.3 

n44.o 

1240.6 

i34i.2 

i445.6 

i553.9 

I666.I 

1782.3 

1902.2 

10 

963.9  1062.8 

n45.6 

1242.3 

i342.9 

1447.4 

1555.8 

1668.  o 

1784.2 

1904.3 

ii 

966.4 

1064.  3 

1147.2 

1243.  9 

i344.6 

i449.2 

1667.6 

l669.  9 

1786.2 

1906.3 

12 

966.9 

io55.9 

n48.8 

1245.6 

i346.3 

i45i.o 

i559.5 

i67i.9 

1788.2 

I9o8.4 

i3 

968.3 

1067.4 

n5o.4 

1247.2 

i348.o 

1462.  8 

i56i.3 

i673.8 

1790.1 

I9IO.^ 

i4 

969.8 

i658.9 

1162.0 

i248.9 

1349.7 

1454.5 

i563.2 

i675.7 

1792.1 

I9I2.4 

i5 

97L2 

1060.4 

n53.6 

1260.5 

i35i.4 

i456.3 

i565.o 

i677.6 

1794.1 

I9l4.4 

16 

972.7 

1062.0 

u55.2 

1262.2 

i353.2 

i458.i 

i566.9 

1796.1 

I9I6.5 

i7 

974.1 

io63.5 

n56.8 

1253.8 

i354.9 

i459.9 

1668.7 

1681.4 

1798.1 

I9I8.5 

18 

975.5 

1066.0 

n58.3 

1255.5 

i356.6 

i46i.6 

1670.6 

i683.3 

1800.  o 

I920.6 

'9 

977-0 

1066.6 

1169.9 

1267.  i 

i358.3 

i463.4 

1672.4 

1686.2 

1802.0 

1922  .6 

20 

978.5 

1068.1 

1161.6 

1268.8 

i36o.i 

i465.2 

1674.3 

i687.2 

i8o4«o  1924.  7 

21 
22 

979-9 
98i.4 

io69.6 

IO7I  .  I 

n63.i 
1164.7 

1260.4 
1262.  i 

i36i.8 
1363.5 

i466.9 
i468.7 

1676.1 
1678.0 

1689.1 
i69i  .0 

1806.9  1926.7 

1807.911928.8 

23 

982.9 

IO72.6 

n66.3 

1263.7 

i365.2 

1470.6 

i579.8 

i692.9 

1809.9 

i93o.8 

24 

984.4 

1074.2 

1167.9 

1266.4 

1367.0 

1472.3 

1681.7 

i694.8 

1811  .9 

i932.9 

26 

985.8 

1076.7 

1169.6 

1267.0 

i368.7 

i474.o 

i583.5 

i696.7 

1813.9 

i935.o 

26 

9^7-3 

1077.2 

1171  .  i 

1268.7 

i37o.4 

1476.  9 

i585.3 

i698.6 

1816.8 

i937.o 

27 

988.8 

1078.7 

1172.7 

1270.3 

l372.I 

i477-7 

1687.2 

1700.6 

1817.8 

i939.o 

28 

99o.3 

1080.3 

H74.3 

1272.1 

i373.9 

i479.5 

1689.1 

1702.6 

1819.8  1941.1 

29 

99i.8 

1081.8 

1176.9 

1273.7 

i375.6 

i48i.3 

i59o.9 

1704.4 

1821.8:1943.1 

3o 

993.2 

io83.3 

1177.6 

1276.4 

i377.4 

i483.i 

i592.7 

1706.3 

1823.8 

i945.2 

3i 

994.7 

1084.8 

1179.1 

1277.1 

i379.o 

i484.9 

i594.6 

1708.2 

1826.8 

I947.2 

32 

996.2 

1086.4 

1180.7 

1278.8 

!38o.8 

i486.  7 

i596.5 

1710.2 

1827.8 

i949.3 

33 

997.6 

1087.  9 

1182.3 

1280.4 

i382.5 

i488.5 

i598.3 

1712.  i 

1829.8 

i95i.3 

34 

999-1 

io89.5 

1183.9 

1282.  i 

1384.2 

1490.3 

1600.2 

1714.0 

i83i.8 

i953.4 

35 

1000.6 

io9i  .0 

ii85.5 

1283.8 

i385.9 

1492.1 

1602.  i 

1716.9 

i833.8 

i955.5 

36 

i  002.  i 

IO92.6 

1187.1 

1286.5 

i387.7 

r493.9 

i6o4.o 

1717.  9 

i835.8 

i957.6 

37 

ioo3.5 

io94.  i 

1188.7 

1287.1 

1389.4 

i495.7 

1606.9 

1719.8 

1837.8 

i959.6 

'38 

1006.  o 

io95.7 

1190.3 

1288.8 

i39i  .2 

1497.6 

1607.7 

1721.7 

i839.8 

1961.7 

39 

1006.  5 

IO97-2 

1191.9 

I290.5 

i392.9 

i499.3 

i6o9.6 

1723.6 

i84i.8 

i963.7 

4o 

1008.  o 

io98.8 

1193.5 

1292.2 

i394.7 

1601  .  i 

1611.6 

1726.6 

i843.8 

ic;65.8 

4  1 

ioo9.4 

noo.  3 

1  196.  i 

I293.8 

i396.4 

i5o2.9 

i6i3.3 

1727.6 

i845.8 

i967.8 

42 

ioio.9 

IIOI  .9 

1196.7 

i295.5 

i398.2 

1604.7 

1616.2 

i729.5 

1847.8 

1969.9 

43 

1012.4 

no3.4 

1198.3 

I297.2 

i399.9 

1606.  5 

1617.  i 

1731.6 

i849.8 

I972-° 

44 

io.i3.9 

1106.0 

1199.9 

1298.9 

1  401.7 

1608.4 

i6i9.o 

i733.4 

1861.8 

i974.  i 

45 

1016.4 

1106.6 

I2OI  .5 

i3oo.5 

i4o3.4 

1610.2 

1620.8 

1735.3 

i853.8 

1976.1 

46 

ioi6.9 

1108.1 

I2O3.  I 

I3O2  .2 

i4o5  .2 

l5l2.O 

1622.7 

1737.2 

1855.8 

1978.2 

47 

1018.4 

no9.6 

1204.7 

i3o3.9 

i4o6.9 

i5i3.8 

1624.6 

i739.2 

1867.8 

1980.3 

.48 

ioi9.9 

IIII  .2 

1206.4 

i3o5.6 

i4o8.7 

i5i5.6 

1626.6 

1741.2 

i859.8 

i"982.4 

49 

IO2I  .4 

1112.7 

1  208.0 

i3o7.3 

i4io.4 

1617.4 

1628.3 

i743.  i 

i86i.fi 

i984.4 

5o 

IO22  .8 

in4..3 

1209.6 

i3o9.o 

l4l2.2 

i5i9.2 

i63o.2 

i745.  i 

1863.8 

i986.5 

Si 

1024.3 

in5.8 

I2II  .2 

i3io.7 

i4i3.9 

1621  .0 

i632.i 

1747.0 

1866.8 

i988.6 

62 

1025.8 

1117.4 

I2I2.9 

i3i2.4 

i4i5.7 

1622.  9 

i634.o 

1749.0 

1867.8 

i99o.7 

53 

I027.3 

1118.9 

i2i4.5 

i3i4.i 

1417.4 

1624.7 

i635.9 

1760.9 

i869.8 

i992.7 

54 

1028.8 

II20.5 

1216.  i 

i3i5.7 

l4l9.2 

1626.6 

i637.7 

1762.9 

1871.8 

i994.8 

55 

io3o.3 

II22.0 

1217.7 

i3i7.4 

l42O.9 

1628.3 

i639.6 

1764.8 

i873.8 

i996.9 

56 

io3i.8 

H23.6 

1219.4 

1422.7 

i53o.2 

i64i.5 

1766.8 

1876.  9 

i999.o 

57 

io33.3 

1126.1 

1221  .O 

i32o.8 

1424.4 

i532.o 

i643.3 

1768.7 

1877.  9 

2OOI  .O 

58 

io34.8 

1126.7 

1222.6 

1322.5 

1426.2 

i533.8 

i645.2 

1760.7 

1879.9 

2O03.  I 

59 

io36.3 

1128.3 

1224.2 

1324.2 

1427.  9 

i535.6 

i647.i 

1762.6 

1882.0 

2oo5.3 

TABLE    X. — REDUCTION   TO   THE    MERIDIAN. 


371 


For  Rate. 

Sec.  32m.  33m. 

34m. 

35m. 

PART  SECOND. 

T. 

Log.  r.  factor. 

" 

" 

" 

" 

in.  s. 

" 

m.   s.\  " 

m.   s 

/' 

S. 

0 

2007.^ 

2i34.6 

2265.6 

2400.6 

5  o 

O.OI 

16   o 

0.61 

26   o 

4.26 

+  3o 

9.9996986 

i  2009.42136.8 

2267.8 

2402.9 

6  o  o.oi 

10 

o.64 

10 

4.37 

29 

7086 

2 

3 

I20H.5  2i38.9 

2013.6  2l4l.I 

2270.0 
2272.2 

24o5.2 
2407.5 

20  O.OI 

3o  0.02 

20 

3o 

0.67 
0.69 

20 

3o 

4.48 
4.60 

28 
27 

7186 
7286 

4 

2015.7  2143.2 

2274.5 

2409.8 

40  O.02 

4o 

0.72 

4o 

4.72 

26 

7387 

5 

2017.8  2i45  .  :. 

2276.7 

24l2.O 

5o  0.02 

5o 

0.76 

5o 

4.83 

26 

7487 

6 

2019.9  2i47.5 

2278.9 

24i4.3 

7  00.02 

17  o 

0.78 

27  o 

4.96 

22 

7688 

7 

2022.0  2l49.  7 

2281.2 

2416.6 

10 

O.02 

10 

0.81 

10 

6.08 

23 

7688 

8 

2024.1  2i5i  .8 

2283.4 

2418.9 

20 

o.o3 

20 

o.84 

20 

5.20 

22 

7789 

9 

2026.2  2i53.9 

2285.6 

2421.2 

3o 

o.o3 

3o 

0.88 

3o 

5.33 

21 

7889 

10  12028.3 

2166.1 

2287.8 

2423.5 

4o 

o.o3 

4o 

o.9i 

4o 

5.46 

2O 

799° 

ii  2o3o.5 

2i58.3 

2290.0 

2425.8 

5o 

o.o4 

5o 

o.95 

5o 

5.6o 

19 

8090 

12 

2032.5 

2i6o.5 

2292.3 

2428.1 

8  o 

o.o4 

18  o 

0.98 

28  o 

5.73 

18 

8191 

i3 

2o34.6 

2162.6 

2294.  5 

2430.4 

IO 

o.o4 

IO 

.02 

10 

5.87 

17 

8291 

i4  2036.7 
i5  2o38.8 

2164.8 
2166.9 

2296  .8 
2209.0 

2432.7 
2435.0 

20 

3o 

o.o5 
o.o5 

20 

3o 

.06 
.09 

20 

3o 

6.01 
6.16 

16 
i5 

8392 
8492 

16 

J7 

2040.9 

2043.0 

2169.  i 
2171.2 

23oi.3 
23o3.6 

2437.3 
2439.6 

4o 
5o 

0.06 
0.06 

4o 
5o 

.13 

.18 

4o 
5o 

6.3o 

6.44 

i4 
i3 

8593 
8693 

18 

2o45.  i 

2i73.4 

23o5.8 

2441.9 

9  ° 

0.06 

I9   0 

.22 

29  o 

6.69 

12 

8794 

J9 

2047.2 

2175.6 

23o8.o 

2444.2 

10 

0.07 

10 

.26 

"\r\ 

10 

6.75 

II 

8894 

onric 

20 
21 
22 
23 
24 
25 

26 
27 
28 
29 

2049.3 
2o5i  .4 
2o53.5 
2055.7 
2057.8 
2059.9 
2062.0 
2064.1 
2066.2 
2068.3 

2177.  8 
2179.9 
2182.  I 
2184.3 
2186.4 
2188.6 
2190.8 
2193.0 
2195.2 
2197.3 

23l0.2 

2312.4 
23!4.7 

23i6.9 
2319.2 

2321.5 

2323.7 
2325.  9 

2328.2 

233o.4 

2446.5 
2448.8 
245i.i 
2453.4 
2455.7 
2458.o 
246o.3 
2462.6 
2464.9 
2467.2 

20 

3o 
4o 
5o 

IO   0 
IO 

20 
3o 
4o 
5o 

II   O 

0.07 
0.08 
0.08 
o.  09 
0.09 

0.  10 
O.II 
O.II 
0.12 

o.i3 

o.i4 

20 

3o 
4o 
5o 
20  o 

10 

20 
3o 
4o 
5o 

21    O 

.  OO 

.35 
.4o 
.44 
•  49 
.54 
.60 
.65 
.70 
.76 
.82 

20 

3o 
4o 
5o 
3o  o 

10 
20 

3o 
4o 
5o 
3i  o 

6  .  90 
7.o6 
7.22 
7.38 
7.55 
7.72 
7.89 
8.06 
8.24 
8.42 
8.61 

I  O 

9 

7 

c 

4 

2 

2 
+  I 

o 

B9(p 
9096 
9I96 

9296 
9397 

9497 

9«9« 

9698 
9799 
9-9999899 
o.ooooooo 

3o 

2O70.4J2I99.  5 

2332.7 

2469.5 

10 

o.i5 

10 

.87 

10 

8.79 

—  I 

O.OOOOIOI 

3i 

2072.6 

22OI  .7 

2334.9 

2471.8 

20 

o.i5 

20 

/ 

.93 

20 

8.98 

2 

O2OI 

32 

2074.7 

2203.9 

2337.2 

2474.2 

3o 

o.  16 

3o 

y 

•  QQ 

3o 

q.  I7 

3 

0302 

33 

2076.8 

2206.  I 

2339.4 

2476.5 

4o 

0.17 

4o 

VV 
2.o6 

4o 

9.37 

4 

04O2 

34 

2078.9 

2208.3 

2341.7 

2478.8 

5o 

0.18 

5o 

2.  12 

5o 

9.67 

c 

o5o2 

35 

2081.0 

2210.5 

2343.9 

2481.1 

12   O 

o.  19 

22   0 

2.  19 

32   0 

9.77 

6 

o6o3 

36 

2083.2 

2212.7 

2346.2 

2483.5 

IO 

O.2O 

IO 

2.25 

10 

9.97 

7 

070^ 

37 

2o85.3 

2214.9 

2348.5 

2485.8 

2O 

0.22 

20 

2.32 

20 

10.  18 

080^ 

38 

2087.4 

2217.  I 

235o.7 

2488.1 

3o 

0.23 

3o 

2.39 

3o 

10.39 

g 

o9o5 

39 

2089.6 

2219.3 

2353.0 

2490.4 

4o 

0.24 

4o 

2.46 

4o 

10.  61 

7 

10 

1006 

4o 

2091.7 

2221  .5 

2355.2 

2492.8 

5o 

O.25 

5o 

2.54 

5o 

10.82 

ii 

1  1  06 

4i 

2093.8 

2223.7 

2357.5 

2495.1 

i3  o 

0.27 

23   0 

2.61 

33  o 

II  .Ol\ 

12 

1206 

42 

2095.9 

2225.9 

2359.7 

2497.4 

IO 

0.28 

IO 

2.69' 

IO 

11.27 

i3 

1  307 

43 

2098.0 

2228.  I 

2361.9 

2499.7 

20 

o.3o 

20 

2.77 

20 

ii.  5o 

i4 

1407 

44 

2IOO.2 

2230.3 

2364.2 

* 

2502.1 

3o 

o.3i 

3o 

2.85- 

3o 

ii.  73 

i5 

1608 

45 

2102.3 

2232.5 

2366.4 

25o4.4 

4o 

o.33 

4o 

2.93 

4o 

ii  .96 

16 

1608 

46 

2  i  o4  .  5 

2234.7 

2368.7 

2506.7 

5o 

o.34 

5o 

3.oi 

5o 

12.  2O 

17 

I7o9 

47 

2io6»6 

2236.9 

2371.0 

2509.0 

i4  o 

o.36 

24  o 

3.io 

34  o 

12.44 

18 

i8o9 

48 

2108.  8 

2239.  I 

2373.3 

2611.4 

IO 

o.38 

IO 

3.i8 

10 

12.69 

J9 

I9IO 

49 

2IIO.Q 

2241.3 

2375.5 

25x3.7 

20 

o.39 

20 

3.27 

20 

12.94 

20 

20IO 

5o 

2Il3.  I 

2243.5 

2377.8 

2616.1 

3o 

0.41 

3o 

3.36 

3o 

10.19 

21 

21  1  I 

5i 

52 

53 

54 

2Il5  .2 
2II7.4 
2119.6 
2I2I.7 

2245.7 
2247.9 

225o.  I 
2252.3 

238o.i 

2382.4 

2384.6 
2386.  9 

2618.4 
2620.8 
2623.1 
2626.4 

4o 
5o 
i5  o 

IO 

o.43 
o.45 
0.47 
0.49 

4o 
5o 
26  o 

10 

3.45 
3.55 
3.64 

3.74 

4o 
5o 
35  o 

IO 

i3.45 
i3.7i 

i3.97 
i4.24 

22 
23 

24 

26 

2211 
23l2 
24l2 

25i3 

55 
56 

57 
58 
59 

2123.8 

2126.0 

2128.1 

2i3o.3 

2132.4 

2254.5 
2256.7 
2258.9 
2261  .  1 

2263.4 

2389.2 

239i.5 
2393.7 
2396.0 
2398.3 

2627.7 
253o.i 
2532.4 
2534.8 
2537.i 

20 

3o 
4o 
5o 
16  o 

0.62 
0.54 
o.56 
o.59 
0.61 

20 

3o 
4o 
5o 
26  o 

3.84 
3.94 
4.o5 
4.i5 
4.26 

20 

3o 
4o 
5o 
36  o 

i4.5i 
14.78 
16.06 
i5.35 
i5.63 

26 

27 

28 

I9 

—  3o 

26i3 
2714 

2814 

29l5 

o.ooo3oi5 

372 


TABLE    X I. — F OR   THE    EQUATION    OF    EQUAL 


Logarithm  of  A. 

Mm.  2h.   3h.   4h.  i  5  h. 

6h. 

7h.   8h. 

9h.  |  10  h. 

lib. 

12  h. 

o  '9  .4109  9  .41  72 

9  .4260 

9.4374 

9.45!5 

9.4685 

9.4884 

9.5n5 

9.5379 

9.568o 

9  .6021 

2 

.4m 

.4i?4 

.4263 

.4378 

.4521 

.469i 

.4892 

.5i23 

.5389 

.569i 

.6o33 

4 

.4n3 

.4177 

.4266 

.4383 

.4526 

.469? 

.4899 

.5i32 

.5398 

.5701 

.6o45 

6 

•  4n4 

.4179 

.4270 

.4387 

.453i 

•  47o4 

.4906 

.5i4o 

.54o8 

.57I2 

.6067 

8 

.4n6 

.4182 

.4273 

.439i 

.4536 

•47io 

.4913 

.5i48 

•  54i7 

.5723 

.6069 

10 

9-4n8 

9.4184 

9.4277 

9.4396 

9.4542 

9.47i6 

9.4921 

9.5i57 

9.5427 

9.5734 

9.6082 

12 

.4120 

•  4i87 

.4280 

•  44oo 

•  4547 

.4723 

.4928 

.5i65 

.5436 

.5745 

.6o94 

i4 

.4121 

.4190 

.4284 

.44o5 

.4552 

.4729 

.4935 

•  5i74 

.5446 

.5756 

.6106 

16 

.4123 

.4i93 

.4288 

.4409 

.4558 

.4735 

.4943 

.5i82 

.5456 

•5767 

.6119 

18 

.4is5 

.4195 

.4291 

•  44i4 

.4563 

.4742 

.4960 

.5i9i 

.5466 

.5778 

.6i3i 

20 

9-4127 

9.4198 

9.4295 

9.4418 

9.4569 

9-4748 

9.4958 

9.5i99 

9.5475 

9.5789 

9.6i44 

22 

.4129 

.4201 

.4299 

.4423 

•  4574 

.4?55 

.4965 

.5208 

.5485 

.58oo 

.6i56 

24 

.4i3i 

.4204 

.4302 

.4427 

.458o 

.4761 

•  4973 

.52I7 

.5495 

.58n 

.6169 

26 

.4i33 

.4207 

.43o6 

.4432 

.4585 

.4768 

•  498o 

.5225 

.55o5 

.5822 

.6182 

28 

•  4i35 

.4209 

.43io 

•  4437 

.459i 

.4774 

.4988 

.5234 

.55x5 

.5834 

.619^! 

3o 

9.4i37 

9.4212 

9.43i4 

9-444i 

9.4597 

9-4781 

9.4996 

9.5243 

9.5525 

9.5845 

9.62O7 

32 

.4i39 

.42i5 

•  43i7 

.4446 

.4602 

.4788 

.5oo3 

.5252 

.5535 

.5856 

•  O2  2C 

34 

•  4i4i 

.4218 

.4321 

.445  1 

.4608 

.4794 

.5oi  i 

.5261 

.5545 

.5868 

.6233 

36 

•  4i44 

.4221 

.4325 

.4456 

.46i4 

.4801 

.5019 

.6269 

.5555 

.5879 

.6246 

38 

.4i46 

.4224 

.4329 

.446o 

.4620 

.4808 

.5o27 

.5278 

.5565 

.589i 

.6259 

4o 

9.4i48 

9.4227 

9.4333 

9.  4465 

9.4625 

9.48i5 

9.5o35 

9.5287 

9.5576 

9  .59O2 

9.6272 

42 

.4i5o 

.4231 

.4337 

.4470 

.463i 

.4821 

.5o42 

.5296 

.5586 

.59i4 

.6285 

44 

.4i52 

.4234 

.434i 

.4475 

.4637 

.4828 

.5o5o 

.53o5 

.5596 

.5926 

.6298 

46 

.4i55 

.4237 

.4345 

.448o 

.4643 

.4835 

.5o58 

.53i5 

.56o6 

.5937 

.63u 

48 

•  4i57 

.4240 

.4349 

.4485 

.4649 

.4842 

.5o66 

.5324 

.56i7 

•  5949 

.6325 

5o 

9.4i59 

9.4243 

9.4353 

9.4490 

9-4655 

9.4849 

9-5o74 

9.5333 

9.5627 

9  .5961 

9.6338 

62 

.4162 

.4246 

.4357 

•  4494 

.466i 

.4856 

.5o82 

.5342 

.5638 

.5973 

.635i 

54 

.4i64 

.425o 

.436i 

.45oo 

.4667 

.4863 

.5091 

.535i 

.5648 

.5985 

.6365 

56 

•  4i67 

.4253 

.4366 

.45o5 

.4673 

.4870 

.5099 

.536i 

.5659 

.5997 

.6378 

58 

.4169 

.4256 

.437o 

.45  10 

.4679 

.4877 

.5io7 

.537o 

.6669 

.  6009 

.6392 

Logarithm  of  A. 

Win. 

13  h. 

14  h. 

15  h.  16  h. 

17  h. 

18  h. 

19  h. 

20  h.  21  h. 

22  h. 

23  h. 

0 

9.6406 

9.6841 

9.7333 

9.7896 

9.8539 

9.9287 

0.0172 

o.  1249 

0.2623 

0.4520 

o.7689 

2 

.6419 

.6856 

•  735i 

.79i5 

.8562 

•  93i4 

.0204 

.  1290 

.2676 

.4601 

.7842 

4 

.6433 

.6872 

.7369 

.7935 

.8585 

.934i 

.0237 

.i33o 

.2729 

.4680 

.8000 

6 

.6447 

.6887 

.7386 

.7955 

.8608 

.9368 

.0270 

.i'37i 

.2783 

.4761 

.8i63 

8. 

.646: 

.6903 

•  74o4 

•7975 

.8632 

.9396 

.o3o3 

.1412 

.2838 

.4842 

.8333 

10 

9.6474 

9.6919 

9.7422 

9-7996 

9.8655 

9.9424 

o.o336 

o.  i45^ 

0.2893 

0.4926 

o.85o8 

12 

.6488 

.6934 

.7440 

.8016 

.8679 

.945i 

.0370 

.1496 

.2949 

.5oio 

.869i 

i4 

.65o2 

.695© 

.7458 

.8o37 

.87o3 

.9479 

.o4o3 

.i538 

.3oo5 

.5097 

.8882 

16 

.65i6 

.6966 

.7476 

.8o58 

.8727 

.9508 

.o437 

.i58i 

.3o63 

.5i84 

.9o8o 

18 

.653o 

.6982 

.7494 

.8078 

.875i 

.9536 

.0472 

.  1623 

.3120 

.5274 

.9288 

20 

9.6545 

9.6998 

9.7512 

9.8o99 

9.8775 

9.9564 

o.o5o6 

o.i667 

o.3i79 

0.5365 

o.95o6 

22 

.6559 

.7014 

.753i 

.8120 

•8799 

.9593 

.o54i 

.1711 

.3238 

.5458 

•  9734 

24 

.6573 

•  7o3o 

•  7549 

.8i4i 

.8824 

.9622 

.o576 

.i755 

.3298 

.5553 

•9975 

26 

.6588 

.7047 

.7568 

.8162 

.8848 

.965i 

.0611 

.1799 

.3359 

.5649 

i  .0228 

28 

.6602 

.7063 

.7586 

.8184 

.8873 

.968o 

.0646 

.i844 

.3420 

.5748 

•  o497 

3o 

9.6616 

9.7079 

9.76o5 

9.82o5 

9.8898 

9-9709 

0.0682 

0.1889 

0.3482 

o.5848 

i.o783 

32 

.663i 

.7096 

.7624 

.8227 

.8923 

.9739 

.0718 

.i935 

.3545 

.595i 

.io89 

34 

.6645 

.7112 

.-7642 

.8248 

.8948 

•9769 

.0754 

.1981 

.36o9 

.6o56 

.i4i6 

36 

.6660 

.7129 

.7661 

.8270 

.8973 

•9798 

.0790 

.2028 

.3674 

.6164 

.I770 

38 

.6675 

•  7i46 

.7680 

.8292 

.8999 

.9829 

.0827 

.2075 

.3739 

.6273 

.2154 

4o 

9.6690 

9.7162 

9.7699 

9.83i.4 

9.9024 

9.9859 

0.0864 

O.2I22 

o.38o5 

0.6386 

r.2573 

42 

.6704 

.7179  .7718 

.8336 

.9060 

.9889 

.0901 

.2I7O 

.3873 

.65oi 

.3o37 

44 

.6719 

.7196 

•7738 

.8358 

.9075 

.9920 

*o939 

.2218 

.394i 

.6619 

.3554 

46 

.6734 

.7213 

.7757 

.838o 

.9101 

.995i 

.o976 

.,2267 

.4oio 

.674o 

.4i4o 

48 

.6749 

.7230 

.7776 

.8402 

•9I27 

.9982 

.ioi5 

.23x6 

,.4o8o 

.6865 

.48i5 

5o 

9.6764 

9.7247 

9«7796 

9.8425 

9.9154 

o.ooiS 

o.io53 

0.2366 

o.4i5i 

0.6993 

i.56i3 

52 

.6779 

.7264 

.78i5 

.8447 

.9180 

,oo44 

.  1092 

.2416 

.4223 

.7124 

.6588 

54 

.6796 

.7281 

.7835 

.8470 

.9206 

.0076 

.n3i 

.2467 

•4297 

.7269 

.7844 

56 

.6810 

.7299 

.7855 

.8493 

.9233 

.0108 

.1170 

.25i8 

.4371 

.7398 

.9610 

58 

.6826 

•  73i6 

.7875 

.85i6  .9260 

,oi4o 

.  1209 

.2670 

.4446 

.754i 

2  .  2627 

ALTITUDES    OF   THE    SUN. 


Logarithm  of  B.                        B  neg. 

Win.  2h.   3  h.   4  h.   5  h. 

6h. 

7h. 

8h. 

9h.   lOh.  |  11  h.  12h. 

o  9.3959 

9.3828 

9.3635 

9.3369 

9.3oio 

9.253o 

9.i874 

9.o943 

8.95o9 

8.6837 

Inf. 

2 

.3955 

.3822 

.3627 

.3358 

.2996 

.a5ii 

.1848 

.o9o6 

.9447 

.6701 

7.243i 

4 

.3952 

,38i7 

.3620 

.3348 

.2982 

.2492 

.1822 

.0867 

.9384 

.656o 

.5453 

6 

.3948 

.38n 

.36i2 

.3337 

.2968 

.2473 

.i796 

.0828 

.9320 

.64i4 

.-7226 

8 

.3944 

.38o6 

.36o4 

.3327 

.2954 

.2454 

.i769 

.0789 

.9254 

.6262 

.8488 

10 

9.394i 

9.38oo 

9.3596 

9.33r6 

9.294o 

9.2434 

9.I742 

9.o749 

8.9i87 

8.6io3 

7-9469 

12 

.3937 

-3794 

.3588 

.33o5 

.2925 

.24i5 

.  i7i5 

.0708 

.9n8 

.5937 

8.0273 

i4 

.3933 

.3789 

.358o 

.3294 

.29II 

.2395 

.1687 

.0667 

.9o48 

.5764 

.o955 

16 

.3929 

.3783 

.3572 

.3283 

.2896 

.2375 

.i659 

.0625 

•8977 

.5583 

•  i547 

18 

.3925 

•3777 

.3564 

.3272 

.2881 

.2355 

.i63o 

.o583 

.89o3 

.5392 

.2071 

20 

9.392i 

9.377i 

9.3555 

9.326i 

9.2866 

9.2334 

9.  1602 

9.o54o 

8.8829 

8.5i92 

8.254i 

22 

.8917 

.3765 

.3547 

.3249 

.285o 

.23i3 

.i573 

.o496 

.8752 

.498i 

.20.67 

24 

.39i3 

.3759 

.3538 

.3238 

.2835 

.2292 

.i543 

.o452 

.8674 

.4758 

.3357 

26 

.39o9 

.3752 

.353o 

.3226 

.28l9 

.227I 

.i5i3 

.o4o6 

.8594 

.4521 

.37i7 

28 

.39o5 

.3746 

.35ai 

.32i4 

.2804 

.225o 

.i483 

.o36o 

.85i2 

.42-70 

.4o5i 

3o 

9.39oo 

9.374o 

9.35i2 

9.32o3 

9.2788 

9.2228 

9.i453 

9.o3i4 

8.8427 

8.4ooi 

8.4363 

32 

.3896 

.3733 

.35o3 

.3i9i 

.2772 

.2206 

.  l422 

.0266 

.834i 

.37i3 

.4657 

34 

.3892 

.3727 

.3494 

.3i78 

.2756 

.2184 

.  i39o 

.0218 

.8253 

.34o  3 

.4932 

36 

.3887 

.3720 

.3485 

.3i66 

.2739 

.2l62 

.i359 

.oi69 

.8162 

.3o67 

.5l92 

38 

.3882 

,37i3 

.3476 

.3i54 

.2723 

,2l4o 

.l327 

.ou9 

.8068 

.27OI 

.544o 

4o 

9.3878 

9-37°7 

9.3467 

9.3i42 

9.27o6 

9.2II7 

9.1294 

9.oo69 

8.7972 

8  ,2299 

8.5675 

42 

.3873 

.37oo 

.3457 

.3i29 

.2689 

.2094 

.  1261 

.0017 

.7873 

.i853 

.5899 

44 

.3868 

.3693 

.3448 

.3n6 

.2672 

.2070 

.1228 

8.9965 

.7772 

.i354 

.6u4 

46 

.3863 

,3686 

.3438 

.3io3 

.2655 

.2047 

.  u94 

.99u 

.7668 

.0-786 

.6320 

48 

.3859 

.3679 

.3429 

.3o9i 

.2638 

.2023 

.  n59 

.9857 

.756o 

.0128 

.65i7 

5o 

9.3854 

9.3672 

9.34i9 

9.3o78 

9.2620 

9.I999 

9.II25 

8.98o2 

8.7449 

7.9348 

8.67o7 

62 

.3849 

.3665 

.34o9 

.3o64 

.2602 

.I974 

.  io89 

.9745 

.7335 

.839i 

.689o 

54 

.3843 

.3657 

.3399 

.3o5i 

.2584 

.  i95o 

.io54 

.9688 

.7217 

•  7i54 

.7067 

56 

.3838 

.365o 

.3389 

.3o38 

.2566 

.  I925 

.  1017 

.963o 

•  7094 

.54o5 

.7237 

58 

.3833 

.3643 

.3379 

.3o24 

.2548 

.  i9oo 

.o98i 

.957o 

.6968 

.24o7 

.7402 

Logarithm  of  B  negative. 

Mm, 

13  h. 

14  h. 

15  h.  16  h. 

17  h. 

18  h.  |  19  h. 

20  h.  |  21  h. 

22  h. 

23  h. 

o 

3-7563 

9.o97i 

9.3i62 

9.4884 

9.6383 

9.7782 

9.9i67 

0.0625 

0.2279 

o.4372 

o.7652 

2 

•7?i8 

.  io57 

.3225 

•  4937 

.643i 

.7827 

.92l3 

.o676 

.2339 

.4455 

.7807 

4 

.7868 

.n4i 

.3287 

.499o 

.6478 

.7873 

.926o 

.o727 

.2401 

.454o 

•7967 

6 

.8oi5 

.  1224 

.335o 

.5o42 

.6526 

.79i9 

.93o7 

•°779 

.2462 

.4625 

.8i33 

8 

.8i58 

.i3o6 

.34ii 

.5o94 

.6573 

.7965 

.9355 

.o83o 

.2524 

.47n 

.83o5 

10 

8.8296 

9.i387 

9.3472 

9.5i46 

9.662i 

9.8on 

9.9402 

0.0882 

0.2587 

0.4799 

0.8483 

12 

.8432 

.i468 

.3533 

.5i97 

.6668 

,8o57 

.9449 

.o935 

.265o 

.4889 

.8667 

14 

.8564 

.i54? 

.3593 

.5248 

.67i5 

.8io3 

•9497 

.o987 

.2714 

.498o 

.8860 

1.6 

.8692 

.1625 

.3653 

.53oo 

.6762 

.8i49 

.9544 

.  io4o 

.2778 

.5072 

.9o6o 

18 

.8818 

.I7o3 

.37i3 

.535i 

.68o9 

.8i95 

.9592 

.io93 

.2843 

.5i65 

.9270 

20 

8.894i 

9.i779 

9-3772 

9.54oi 

9.6856 

9.824i 

9.964o 

o.n46 

O.29O9 

0.5261 

o.9489 

22 

•  9o62 

.1855 

.383i 

.5452 

.69o3 

.828-7 

.9687 

.  1  200 

.2975 

.5358 

.97i9 

24 

.9i8o 

.i93o 

.3889 

.55o2 

.6949 

.8333 

.9735 

.1253 

.3o4i 

.5457 

.996i 

26 

.9295 

.2004 

.3947 

.5553 

.6996 

.8379 

•  9784 

.i3o8 

.3io9 

.5557 

i  .0216 

28 

.94o8 

.2O78 

,4oo5 

.56o3 

.7o43 

.8425 

.9832 

.i362 

•3i77 

.566o 

.0487 

3o 

8.95i9 

9.2i5o 

9  .4062 

9.5653 

9.7o89 

9.847i 

9.988o 

o.i4i7 

0.3245 

0.5764 

1.0774 

32 

.9627 

.2222 

•  4n9 

.5702 

.7i36 

.85i7 

•9929 

.1472 

.33i5 

.587i 

.1081 

34 

.9734 

.2293 

.4i75 

.5752 

.•7182 

.8563 

•9977 

.  1527 

.3385 

.5979 

.i4o9 

36 

.9839 

.2364 

.4232 

.58oi 

.7228 

.86o9 

0.0026 

.i582 

.3456 

.6o9o 

.1764 

38 

.9942 

.2434 

.4288 

.585o 

.7275 

.8655 

.oo75 

.i638 

.3527 

.6204 

.2l49 

4o 

9.oo43 

9  .25o3 

9.4343 

9  .59oo 

9.732I 

9.87oi 

0.0124 

o.i695 

o.3599 

o.63i9 

i.2569 

42 

.0142 

.257I 

•  4399 

.5948 

.7367 

.8748 

.oi73 

.  i75i 

.3673 

.6438 

.3o33 

44 

.0240 

.2639 

.4454 

.5997 

•  74i3 

•  8794 

.O223 

.1808 

.3747 

.6559 

.3552 

46 

.o336 

.2706 

,45o9 

.6o46 

.7459 

.8840 

.0272 

.1866 

.3822 

.6684 

.4x38 

48 

.o43i 

.2773 

.4563 

.6o94 

.75o5 

.8887 

.0322 

.I924 

.3897 

.6811 

.48:4 

5o 

9.o524 

9.2839 

9.4617 

9.6i43 

9.7552 

9.8933 

o.o372 

O.I982 

o.3974 

o.6942 

i.56i2 

52 

.0616 

.29o5 

.4671 

.6i9i 

.7598 

.898o 

.O422 

.2040 

•  4o52 

.7o76 

.6587 

54 

.o7o7 

.2970 

.4725 

.6239 

.7644 

.9026 

.0473 

.2O99 

•  4i3o 

•  72i4 

.7843 

56 

.o796 

.3o34 

.4778 

.628-7 

.769o 

.0.073 

.o523 

.2l59 

.4210 

.7355 

.96io 

58 

.0884 

.3o98 

.483i 

.6335 

.7736 

,9I20 

.0574 

.22I9 

.429I 

.75oi 

2.2627 

374  TABLE  XI I. — L ENGTH  OF  A  DEGREE   OF  LONG.  AND  LAT. 


Geograph. 
Latitude. 

Angle  of 
Vertical. 

Diff. 

Logarithm  of 
Earth's  Radius. 

Diff. 

Deg.  of  Meridian. 
English  Feet. 

Diff. 

Deg.  of  Parallel. 
English  Feet. 

Diff. 

0      0 
I       0 
2       0 
3      0 
4      0 

5     o 
6     o 
V      o 
8     o 

9     ° 

10       0 
II        0 
12        0 

i3     o 
i4    o 
i5     o 
16     o 
17     o 
18     o 

0      O.OO 
o  24.02 
o  48.02 
i   ii.  95 

i   35.8o 
i  59.54 

2    23.12 

2  46.54 
3     9.76 
3  32.  74 
3  55.47 
4  17.92 
4  4o.o6 
5     i.85 
5  23.28 
5  44.33 
6     4.95 
6  25.i4 
6  44.86 

24.02 
24.00 
23.  93 
23.85 
23.  74 
23.58 

23.42 
23.22 
22.  98 
22.  73 

22.45 

22.  l4 

21-79 

21.43 

21.  o5 
20.62 

20.  I9 

19.72 
19.23 

o.ooooooo 
9.9999996 
9982 
9961 

9891 
9.9999843 
9786 
9721 
9648 
9666 
9476 

9-9999377 
9271 
9167 
9o35 
89o5 
8768 
9.9998624 

4 
i4 

21 

3i 
39 

48 

57 
65 

73 
82 

9° 
99 
106 

122 

i3o 
i37 
1  44 
162 

362748.33 
749.43 
762.  75 
768.28 
766.00 
776.  9i 
362788.01 
802.27 
818.68 
837.22 
867.86 
880.  59 
3629o5.37 
932.i8 
96o.98 
991.74 

o58.99 
363o95.4o 

I  .  10 

3.32 
5.53 

7-72 
9.91 

12.  IO 

14.26 

16.41 
i8.54 
20.64 

22.  73 

24.  78 
26.81 
28.80 
3o.76 
32.68 
34.57 
36.  4i 

38.  3T 

365i85.7l 
5i3o.47 
4964.74 
4688.67 
43o2.o5 
3806.29 
363i98.43 
248i.64 
I  655.  i  3 
0719.  i3 
369673.92 
8619.79 
367267.07 
5886.13 
44o7.  36 
2821  .  19 
1128  .07 
349328.60 
347422.98 

55.24 
i65.73 
276.  17 
386.52 
496.76 
606.86 
716.79 
826.61 
936.00 
1046.21 
n54.i3 
1262.72 
1370.94 

1478.77 
1686.17 
i693.  12 
1799.67 
1906.62 
2010.91 

i9     o 

2O        0 
21        O 
22        0 
23        0 

24     o 
26     o 
26     o 

7     4-o9 
7  22.80 
7  4o.99 
7  58.  61 
8  16.66 
8  32.io 
8  47.93 
9     3.i2 

18.71 
18.19 
17.62 
17.06 

16.44 

i5.83 
16.19 

14.53 

8472 
83i4 
8i49 
7977 
7799 
9-99976i4 
7424 
7228 

i58 
i65 

I72 

178 

i85 

I9O 

i96 

201 

i33.6i 
I73.57 

2l5.23 

268.66 
3o3.49 
363349.96 
397.93 
447-34 

39.96 
4i.66 
43.33 
44.93 
46.47 
47.97 
49.41 
60.70 

54i2  .07 
3296.  36 
1076  .45 
338762.98 
6326.62 
333798.08 
1168.  10 
328437.44 

2116.71 
2219.91 
2323.47 
2426.36 
2628.64 
2629.98 
2730.66 
2830.55 

27     o 
28     o 

29        O 

3o     o 

10 
20 

3o 
4o 
5o 
3i      o 

10 

9  17.66 
9  3i.5o 
9  44.66 
9  67.12 

69.  12 
10        I  .  I  I 
3.07 
5.02 

6.94 

10     8.85 
10.73 

i3.85 
i3.i6 

12.46 
2.OO 

•99 
.96 
.96 
.92 
.91 
.88 
8fi 

7027 
6820 
6608 
9.9996392 
6355 
6319 
6282 
6246 
6208 
9.9996171 
6i34 

207 
212 

216 

37 

36 
37 
37 
37 
3? 
37 
38 

498.i3 
660.24 
6o3.6o 
363658.i4 
667.34 
676.68 
685.84 
695.i4 
7o4.46 
3637i3.8i 

723.2O 

C 

62.  1  1 
53.36 
54.54 
9.20 
9.24 
9.26 
9.3o 
9.32 

9.35 
-9.39 

6606.89 
2677  -27 
319649.44 
316624.29 
6994.03 
546i  .  10 
4926  .  5o 
4387.23 
3846.3i 

2766.48 

2929.62 
3o27.83 
3i25.i5 
53o.26 
532.  93 
535.60 
538.27 
540.92 
643.69 
546.24 
6/8   88 

20 

3o 

4o 
5o 

32        0 
10 
20 

3o 
4o 
5o 
33     o 

IO 

20 
3o 
4o 

12.  59 

14.44 
16.26 
10.66 

10  I9.84 
21.60 
23.34 
26.06 
26.76 
28.43 
10  3o.o8 
31.71 
33.32 
34.91 
36.48 

.85 
.82 
.80 
•78 
-76 
•74 

.70 
.68 
.65 
.63 
.61 
.69 

6096 
6069 
6021 
5984 
9.9996946 
6908 
6870 
5832 
5794 
5755 

9-99957I7 
6678 
564o 
6601 
6662 

37 
38 

37 
38 
38 
38 
38 
38 
39 
38 
39 
38 

732.6i 

742.o5 
761.62 

76l.02 

36377o.54 
780.  10 
789.68 

799-29 
808.92 
818.68 
363828.27 
837.98 
847.72 
857.48 
867.26 

9-44 

9-47 
9.60 
9.62 
9.66 
9.68 
9.61 
9.63 
9.66 
9.69 
9.71 
9.74 
9.76 
9.78 
o  81 

2207.60 
1666.08 
noi  .  91 
o545.ii 
309986.68 
9423.63 
8858.  96 
8291.66 
7721.76 
7149.26 
3o6574.i4 
6996.43 
54i6.i3 
4833.24 
4247.77 

55i.52 

664.17 
556.8o 
669.43 
662.06 
564.68 
667.29 
669.90 
672.61 
676.11 
677.71 
58o.3o 
682.89 

585.47 
588  06 

5o 

38.  o3 

K0 

5523 

9 

877.07 

9C  / 

3669.72 

34     o 

IO 

20 
3o 
4o 
5o 
35     o 

10  39.55 
4i.o6 
42.54 
44-00 
45.44 
46.86 
10  48.26 

48 
.46 
.44 
.42 
.39 

9.9995484 
5445 
54o6 
5367 
5327 
6288 

39 
39 
39 
4o 
39 
4o 

363886-91 
896.  77 
906.66 
916.66 
9.26.48 
936.43 
363946.4o 

.04 

9.86 
9.88 
9.90 
9.93 
9.96 

9-97 

3o3o69  .  10 
2476  .91 
1880.16 
1281.84 
0680.97 
0077.66 
299471  .60 

693.  19 
595.75 
698.32 
600.87 
6o3.42 
606  .96 

ANGLE  OF  THE  VERTICAL  AND  LOG.  OF  EARTH'S  RADIUS.  375 


Geograph. 
Latitude. 

Angle  of 
Vertical. 

Diff. 

Logarithm  of 
Earth's  Radius. 

Diff. 

Deg.  of  Meridian. 
English  Feet. 

Diff. 

Deg.  of  Parallel. 
English  Feet. 

Diff. 

35     o 

10 
20 

3o 
4o 
5o 
36     o 

10 

20 
3o 

10   48.a5 
49-63 
60.98 
62.81 
53.62 
54.90 
10    56.16 
57.41 

58.63 
69.82 

.38 
.35 
.33 
.3i 

.28 
.26 

.25 
.22 

9.9996248 
6208 
6169 
6129 
6089 

5o49 
9.9995oo9 
4969 
4929 

4888 

4o 
89 

4o 
4o 
4o 
4o 
4o 
4o 
4i 

Art 

363946.4o 
966.89 
966.41 
976.44 
986.  49 
006.67 
354006.67 
016.78 
026.  9i 
037.07 

9.99 

IO.O2 

10.  06 

10.  08 

10.10 
10.  II 

10.  i3 
10.  16 

299471  .60 
8868.09 
8262.06 
7688.49 
7O22.4O 

64o3.8o 
296782.67 
6169.04 
4682.90 
8904.27 

608.  5i 
6n.o3 
618.67 
616.09 
618.60 
621.18 
628.68 
626.14 
628.63 

5o 
87     o 

10 

II        I.OO 
2.l5 

II    3.28 
4.39 

.15 

.13 

.  ii 

08 

4848 
48o7 
9.9994767 
4726 

4i 
4o 
4i 

An 

047.24 
067.43 
354067.64 
077.86 

10.  19 

IO.2I 
IO.22 

8278.14 
2689.62 
292008.42 
i364.84 

633.62 
636.io 
638.58 
64  1  06 

20 

Bo 

4o 
5o 
38     o 

IO 
20 

3o 
4o 
5o 
39     o 

IO 

20 

3o 

5.47 
6.54 
7.68 
8.59 
ii     9.59 
10.66 
ii.  5i 
12.44 
i3.34 

l4-22 

ii   i  5.  08 
16.92 

16.78 

17.62 

.07 
.04 

.01 

.00 

:95 

.93 
.90 

.88 
.86 
.84 
.81 

-79 

4686 
4645 
46o4 
4563 
9.9994622 
448  1 
444o 
4399 
4358 
43i7 

9'"94234 
4i93 
4162 

4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 
4i 

42 

4i 
4i 

088.10 
o98.36 
io8.64 
n8.93 
364i29.24 
i39.56 
149.90 
160.26 
170.62 
181.00 
364i9i.4o 

2OI  .80 
212.22 
222.66 

10.26 
IO.28 
10.29 

10.  81 
10.82 
10.84 
10.35 
10.87 
10.38 
10.  4o 
10.  4o 
10.42 
10.44 

0728.79 
0080.28 
289434.3o 
8786.86 
288i34.  97 
748i.63 
6826.86 
6i67.64 
6606.  99 
4843.  93 
284i78.44 
35io.54 
2840.28 

2l67.52 

643.5i 
645.98 
648.44 
660.89 
653.34 
655.  78 
658.21 
660.66 
663.06 
665.49 
667.90 
€70.81 
672.71 

/?  r  T  T 

4o 
5o 
4o     o 

IO 

20 

3o 
4o 
5o 
4i    o 

10 

20 
3o 
4o 
5o 

4s    o 

IO 

20 
3o 
4o 
5o 
43     o 

IO 
20 

3o 
4o 
5o 

44    o 

10 
20 

3o 
4o 
5o 
45     o 

18.29 
19.04 
II  19.76 

20.46 
21.  13 
21.79 
22.42 
23.02 

ii   23.  61 

24.17 
24.70 
26.22 
26.71 
26.18 
ii   26.62 
27.04 
27.44 
27.82 
28.17 
28.60 
ii   28.80 
29.08 
29.34 
29.68 

29.79 
29.98 
ii   3o.i4 
80.29 
3o.4i 
3o.5o 
80.67 
80.62 
3o.65 

.72 
.70 
.67 
.66 
.63 
.60 
.69 
.56 
.53 
.62 
.49 
.47 
.44 
.42 
.4o 
.38 
.35 
.33 
.3o 
.28 
.26 
.24 

.21 

'11 

.16 

.15 
.  12 
.09 
.07 

.06 
.o3 

4no 
4o69 
9.9994027 
3986 
8944 

3902 
386o 
8819 
9.9998777 
3735 
3693 
365i 
36o9 
3567 
9.9993525 
3483 
344i 
3399 
3357 
33i5 

3280 
8188 
3i46 
3io4 
8062 
9.9993oi9 

2977 
2935 

2892 
2850 

2808 

9.9992766 

42 
4i 

42 

42 
4i 

42 
42 

4i 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

43 

42 
42 
42 
42 

43 

42 
42 

43 

42 
42 
42 

233.11 
243.57 
364254.04 
264.52 
276.01 

286.  5i 
296.08 
3o6.55 
864317.08 
327.62 
338.17 
348.  73 
359.3o 
869.87 
36438o.45 
891  .o4 
4oi  .64 
412.24 
422.86 
433.46 
364444.o8 
454.70 
465.33 
475.96 
486.  69 
497.23 
864607.87 
618.62 
629.  16 
539.8i 
55o.46 
661.12 
864671.77 

io.45 
io.46 
10.47 
10.48 
10.49 
10.  5o 
10.62 
10.62 
10.53 
10.54 
10.55 
10.56 
10.67 
10.67 
10.68 
10.69 
10.60 
10.  60 
10.  61 
10.  61 
10.62 
10.62 
10.63 
10.63 
10.63 
io.64 
io.64 
10.65 
10.64 
10.65 
10.65 
10.66 
10.65 

i492.4i 
o8i4.9i 

280135.  01 

279452.75 
8768.io 
8081.09 
739i.7i 
6699.  97 
276006.  89 
53o9.46 
4610.68 
8909.66 
8206.18 
2600.86 

27I792.28 
1081.89 
0869.19 
269654.  I9 

8936.9o 
82  i  7.  32 
26-7496.46 
6771.31 
6o44>9° 
6816.22 
4686.29 
8862.10 
263116.67 
2879.00 
1689.09 
0896.96 
0162.60 
269406.08 
268667.26 

070  .  I  I 
677.50 
679.90 
682.26 

684.65 
687.01 
689.38 
691  .74 
694.08 
696.43 
698.78 

7OI  .  12 
703.43 
706.77 
708.08 
710.89 
7I2.70 
7l5.OO 
717.  29 

7i9.58 
721.87 
724.14 
726.41 
728.68 
780.98 
733.i9 
735.43 
787.67 
739.9i 
742.18 
744-36 
746.67 
748.78 

376   TABLE    XI I. — L ENGTH   OF  A  DEGREE  OF  LONG.  AND  LAT. 


Geograph 
Latitude. 

Angle  of 
Vertical. 

Diff. 

Logarithm  of 
Earth's  Radius. 

Diff. 

Deg.  of  Meridian. 
English  Feet. 

Diff. 

Deg.  of  Parallel. 
English  Feet. 

Diff. 

0               ' 

/                       // 

45     o 

IO 
20 

3o 

4o 

ii   3o.65 
3o.65 
3o.63 
3o.58 
3o.5i 

.00 
.02 

.06 

.07 

9.9992766 

2723 

2681 
2689 
2696 

43 
42 

42 

43 

/o 

864671  .77 

682.42 
593.o8 
6o3.73 
6i4.38 

10.65 
10.66 
10.65 
10.65 
Tn  AE; 

268667.25 
7906.26 
7168.08 
6897.  71 

564o.i  5 

•760.99 

753.i8 
766.87 
767.  56 

5o 

80.42 

.09 

2554 

42 

626.08 

4880.42 

709  .  70 

46     o 

10 
20 

3o 
4o 
5o 

ii   3o.3i 
3o.i7 
3o.oi 
29.82 
29.61 
29.88 

.  II 

"  .16 
.19 

.21 
.23 
ofi 

9.9992612 

2470 
2427 

2385 

2843 

23oo 

42 

42 

43 

42 
42 

43 

/o 

364635.68 
646.33 
656.  98 
667.62 
678.26 
688.  9o 

io.65 
10.65 
10.65 
10.64 
io.64 
10.64 

264118.  5i 
3354.43 
2688.20 
1819.82 
1049.29 
0276.62 

761  .91 

764.08 
766.23 
768.  38 
770.53 
772.67 

47     o 

10 

20 
3o 
4o 
5o 

48     o 

10 
20 

3o 
4o 
5o 
49     o 

10 
20 

3o 
4o 
5o 
5o     o 

10 
20 

3o 
4o 
5o 
5i     o 

10 

20 
3o 

n   29.12 
28.86 
28.54 
28.22 
27.87 
27.60 
ii   27.10 
26.69 
26.24 
26.78 
26.29 
24.78 
ii   24.24 
28.69 
28.11 
22.60 
21.87 

21.22 
II     2O.55 

19.86 

19.  i3 
18.89 
17.68 
16.84 
ii    16.02 
16.19 
i4.33 
i3.45 

.27 
.32 

.35 
.37 
.4o 

.*45 
.46 
-49 

!54 

.55 
.58 
.61 
.63 
.65 
-67 
•7° 
•72 
•74 
•76 

•79 
.82 
.83 
.86 

.88 

9.9992268 
2216 

2I74 
2132 
2089 
2047 
9.9992006 
1968 
1921 
l879 

i837 
1796 
9.9991762 
1711 
1669 
1627 
1686 
1  544 
9.9991602 
i46o 
1419 
i377 
i335 
1294 
9.9991262 

I2II 
1170 
1128 

42 
42 
42 

43 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

4i 

42 
42 
42 

41 

42 
42 

4 

42 

4i 
41 

42 

/r 

364699.54 
710.17 
720.80 
781.42 
742.08 
762.66 
364763.25 
773.85 
784.45 
796.04 
806.62 
816.19 
864826.76 
837.3i 
847.86 
858.  4o 
868.98 
879.45 
864889.96 
900.46 
910.96 
921.43 
981.90 
942.86 
364962.80 
968.28 
978.66 
984.06 

10.63 
10.63 
10.62 
10.  61 
10.62 
10.60 
10.  60 
10.  60 
10.69 
10.58 
10.67 
10.56 
10.56 
10.55 
10.54 
10.53 
10.62 
10.  61 
10.  5o 
10.49 
io.48 
10.47 
10.46 
io.44 
io.43 
10.42 
10  .41 

IO    39 

249601.81 
8724.88 
7945.83 
7164.66 
6881.89 
6696  .01 
2448o8.54 
4018.98 
8227.34 
2433.62 
i637.84 
0889  .99 
240040.09 
239238.i4 
8434.i5 

7628  .  12 
6820.07 
6009  .99 
286197.90 

4383.8o 
8667.70 
2749  .60 
1929  .  62 
1107.46 
280283.42 
229467.42 
8629.45 
7799.54 

776.98 
779.o5 
78i.i7 
783.27 
785.38 
787.47 
789.66 
-791.64 
798.72 
796.  78 
797-85 
799.90 
801.96 
808.99 
806.  o3 
808.  o5 
810.08 
812.09 
814.10 
816.10 
818.10 
820.08 
822.06 
824.04 
826.00 
827.97 
829.91 

4o 
5o 
62     o 

10 

20 
3o 
4o 
5o 
53     o 

10 

12.55 
11.62 
ii   10.67 
9.70 
8.71 

6^60 
ii     4.5i 
3.4o 

.90 
.93 
.96 
•97 
•99 

.02 
.03 
.06 
.09 
.11 

1087 

io46 
9.9991006 
0963 
0922 
0881 
o84o 
0800 
9.9990769 
0718 

41 

4i 

42 

4i 
41 
41 

4o 
41 

4i 

/T 

994.45 
365oo4.83 
365oi5.2o 
O25.55 
o35.88 
o46  .20 
o56.5i 
066.79 
866077.06 
087.82 

10.88 
10.87 
10.35 
10.33 
10.82 
10.81 
10.28 
10.27 
10.26 

6967  .67 
6188.87 
226298.18 
446o.47 
8620  .90 
2779.41 
1936.01 
1090.  72 
220248.54 
219394.47 

83i  .87 
833.8o 
835.  74 
837.66 
839.57 

84i.49 
843.4o 
845.29 
847.18 
849.07 

QCn         0 

20 

2.27 

5 

0677 

41 

An 

097.66 

8543.54 

ofjr,    OT 

3o 
4o 
5o 
54     o 

IO 
20 

3o 
4o 
5o 
55     o 

I  .  12 

10  69.94 
58.  74 
10  67.62 
66.28 
55.02 
53.73 
62.42 
61.09 
10  49.74 

.18 
.20 
.22 
.24 
.26 
.29 

.33 

.35 

0687 
0696 
o556 
9.9990616 

o435 

o355 
o3i5 
9.9990276 

4O 

41 

4o 
4i 
4o 
4o 
4o 
4o 
4o 
4o 

107.78 
117.98 
128.17 
365i38.34 

148.49 
168.61 
168.72 
178.81 
188.88 
866198.98 

IO.2O 
10.19 

10.  17 
10.  16 

IO.I2 
10.  II 

10.09 
10.07 
10.06 

7690.78 
6836.o6 
5979.53 
216121  .  16 
4260.96 
3398.90 

2535.o4 
1669.86 
210801  .85 
209982.66 

864.67 

856.53 
858.  37 
860.21 
862.06 
863.86 
866.69 
867.60 
869.80 

ANGLE  OF  THE  VERTICAL  AND  LOG.  OF  EARTH'S  RADIUS.  377 


Geograph 
Latitude 

Angle  of 
Vertical. 

Diff. 

Logarithm  of 
Earth's  Radius 

Diff. 

Deg.  of  Meridian 
English  Feet. 

Diff. 

Deg.  of  Parallel 
English  Feet. 

Diff. 

55     o 

10  49.74 

•5Q 

9.9990276 

4o 

366198.93 

209982.55 

0 

10 

48.36 

.  OC 

0286 

208.96 

I  O  •  OC 

9061  .44 

071  .  I  I 

20 

46.97 

.89 

0196 

4o 

218.97 

IO.OI 

8l88.56 

872.88 

3o 

45.55 

.42 

oi55 

4o 

228.96 

9.99 

73l3.8o 

874.67 

4o 
5o 

44.ii 
42.66 

•  44 
.46 

/ 

0116 
0076 

89 

4o 

288.92 

248.86 

9.96 
9.94 

/                 V 

6437.44 

5559.23 

876.45 
878.21 

56     o 

10  4i.i6 

•  49 

CT 

9.9990087 

?9 

866268.78 

9.92 

204679.26 

879.97 
00T      *' 

10 
20 

3o 

4o 
5o 

39.65 
38.  i3 
36.58 
35.01 
33.  4i 

*  J  1 

.52 

.55 
.67 
.60 

r> 

9.9989998 
9968 
9919 

9880 
9841 

4o 
39 
39 

'9 

268.68 
278.66 
288.40 
298.28 
808.  o3 

9.90 

9.87 

9.86 
9.88 
9.80 

2914.07 

2028.87 
n4i.93 
200268.28 

ool  .  72 
883.  47 
886.20 
886.94 
888.65 

57     o 

10 
20 

3o 
4o 
5o 

10  81.80 
80.16 
28.60 
26.88 
25.i3 
28.  4o 

.61 
.64 
.66 
.67 
.70 
.78 

/ 

9.9989802 
9764 
9726 
9686 
9648 
9610 

9 

89 
39 

38 
38 

866817.80 
827.66 
887.29 
346.99 
356.66 
366.3i 

9-77 
9.76 
9.73 
9.70 
9.67 
9.66 

i99362  ,9o 
8470.82 
7677.04 
6681.67 
6784.42 
4885.58 

890.88 
892.08 
898.78 
895.47 
897.  i5 
898.84 

58     o 
10 
20 
3o 

10  21.66 
i9.9o 
i8.ii 
i6.3i 

•74 
.76 

•79 
.80 

BQ 

9.9989671 
9533 
9495 

9467 

Q 

38 
38 

QQ 

365375.93 
385.53 
896.  10 
4o4.64 

9.62 
9.60 
9.67 
9.64 

I93985.o8 
8082.91 
2179.09 
1273.63 

900.60 
902.  17 
903.82 
906.46 

4o 
5o 

i4.48 
12.63 

•  O  O 

.85 
fifi 

94i9 
9882 

OO 

37 

QQ 

4i4.i6 
423.65 

Q  •  D2 

9.49 

n     /  ^ 

0866.62 

189467.78 

907.11 

908.74 

_      Q  C 

69     o 

IO 
20 

3o 

o  10.77 
8.88 
6.97 
5.o4 

•  oo 

.89 

^3 

/? 

9.9989344 
9807 
9269 
9282 

OO 

37 

38 

365433.10 
442.53 
45i.93 
46i.3o 

9  .40 

9.43 
9.40 
9.37 

188647.48 
7635.45 
6721.86 
6806.67 

9IO.  OD 

911.98 
918.69 

916.  I9 

4o 
5o 
60     o 

8.08 
i  .  1  1 

9     69.  12 

.96 

•97 

•99 

o   ^A 

9196 
9168 
9.9989121 

37 

470.64 

479  -95 
365489.23 

9.34 

9.3i 

9.28 

4889.90 
8971.53 
i83o5i.59 

916.77 

918.87 

9I9-9^ 

61     o 
62     o 
63     o 
64     o 
65     o 
66     o 
67     o 
68     o 
69     o 

9  46.74 
9  33.65 
9   19.86 
9     5.36 
8  60.21 
8  34.4o 
8   17.97 
8     0.92 
7  43.29 

-5  •  OO 

3.09 

3.80 
4.49 

6.16 
6.81 
6.43 
7.06 
7.68 

8     9T 

8902 

8688 

8479 
8276 
8077 
9.9987884 
7697 
7617 
7842 

219 
214 
209 

204 

199 

198 

187 

180 

I75 

ifift 

544.25 
698.10 
660.70 
702  .00 
761.94 
866800.44 
847.45 
892.91 
936.77 

3  C)  •  O2 

53.85 
62.60 
5i.3o 

49.94 
48.  5o 
47.01 
45.46 
i3.86 

171892.  10 
i6623i.84 
160620.21 
164768.95 
148949.83 
148094.68 
187195.14 
i3i253.i8 

6607.  i5 
6660.26 
6711.  63 
6761  .26 

6809.  12 

5855.20 
6899.49 
6941  .96 

\  y-A  fi  r»        R  f 

70     o 
71      o 
72      o 
78     o 
74     o 
76     o 
76     o 

77     o 

78      o 

79     ° 
80     o 
81     o 
82     o 
83     o 
84     o 
85     o 
86     o 
87     o 
88     o 
89     o 
90     o 

7  26.08 
7     6.33 
6  47.06 
6  27.28 
6     7.08 
5  46.33 
5  26.20 
5     3.67 

4  41.77 
4  19.53 
3  66.96 
3  34.10 
3   10.98 
2  47-63 

2    24.07 

2     o.33 
i  36.44 
i   12.43 
o  48.34 
o  24.18 
o     o.oo 

O  *  Zi  I 

8.75 

9-27 
9.78 
20.26 
20.  7o 

21.  l3 

21.53 
21.90 

22.24 
22.67 

22.86 
23.12 

23.35 
23.56 
23.  74 
28.  89 
24.01 
24.09 
24.16 
24.18 

7i74 
7018 
9.9986869 
67i3 
6573 
644  1 
6817 
6201 
9.9986098 
6998 
6901 
58i8 
5743 
6676 
9.9986619 
6670 
6680 
5498 
5476 
5463 
9.9986468 

1  UO 

161 
1  54 
1  46 

132 

124 
116 
1  08 

100 

92 

83 

75 
67 

57 
49 
4o 
82 

22 
13 

5 

978.97 
866019.45 
366o58.i7 
096.08 
i3o.i4 
i63.3o 
194.61 
228.74 
866260.96 
276.18 
299.21 
820.19 
339.02 
355.69 
866870.19 
882.49 
892.67 
4oo.43 
4o6.o4 
409.42 
366410.54 

[.2  •  2O 

4o.48 
38.  72 
86.91 
35.o6 
33.i6 
81.21 
29.28 
27.22 
5.i7 
8.08 
0.98 
8.83 
6.67 
4.5o 

2.30 

0.08 

7.86 

5.6i 

3.38 

I  .  12 

126270.67 
119249.14 
113190.78 
107097.82 
100970.66 
94812.70 
88626.82 
82410.44 
76x69.97 
69906.87 
68620.07 
573i4.53 
60991  .1-7 
44661.98 
88298.98 
81988.97 
26669.08 
19176.27 
12787.49 
6394.74 
oooo.  oo 

0902  .  oi 

6O2I    .43 

6o58.36 
6098.46 
6126.66 
6167.96 
6187.88 
6214.88 
6240.47 
6264.10 
6286.80 
63o5.54 
6323.36 
6889.19 
6353.o5 
6364.  96 
6874.89 
382.  81 
888.78 
892.76 
394.74 

378            TABLE    XIII. 

Augmentation  of  the  Moon's  Semi-diameter,  on 
account  of  her  apparent  Altitude. 

TABLE    XIV. 

Reduction  of  the  Moon's  equatorial  Parallax. 

App. 
Alt. 

Horizontal  Semi-diameter. 

i 

_3 

Moon's  Equatorial 
Parallax. 

(Latitude. 

Moon's  Equatorial 
Parallax. 

1430 

15  0 

1530 

16  0  1630 

17  0 

53' 

[57' 

|61' 

53' 

57' 

61' 

o 

O.IO 

0.12 

o.i3 

o.  i4 

o.i5 

o.  17 

0 

0 

o.oo 

o.oo 

0.00 

45 

5.29 

5.69 

6.o9 

2 

o.58 

0.62 

0.66 

0.71 

0.76 

0.81 

I 

o.oo 

o.oo 

o.oo 

46 

5.48 

5.89 

6.3o 

4 

i.o5 

I.  12 

1.20 

1.28 

i.37 

i.46 

2 

O.OI 

0.  01 

O.OI 

4? 

5.66 

6.09 

6.52 

6 

i.5i 

1.62 

I.74 

1.86 

1.98 

2.10 

3 

o.o3 

o.o3 

o.o3 

48 

5.85 

6.29 

6.73 

8 

1.98 

2.  12 

2.27 

2.42 

2.58 

2.75 

4 

o.o5 

0.06 

0.06 

49 

6.o3 

6.49 

6.94 

10 

2.44 

2.62 

2.80 

2.99 

3.i8 

3.39 

5 

0.08 

0.09 

0.09 

5o 

6.22 

6.69 

7.i5 

12 

2.90 

3.  II 

3.33 

3.56 

3.78 

4.02 

6 

O.  12 

0.12 

o.i3 

5i 

6.4o 

6.88 

7.36 

i4 

3.36 

3.6i 

3.86 

4.  ii 

4.37 

4.66 

7 

0.16 

O.I7 

0.18 

52 

6.58 

7.08 

7.57 

16 

3.82 

4.  10 

4.38 

4.67 

4-97 

5.28 

8 

O.2O 

O.22 

0.24 

53 

6.76 

7.27 

7.78 

18 

4.28 

4.58 

4.89 

5.22 

5.56 

5.90 

9 

0.26 

0.28 

o.3o 

54 

6.94 

7.46 

7-98 

20 

4.72 

5.o6 

5.4i 

5.76 

6.i4 

6.52 

10 

0.32 

o.34 

o.37 

55 

7.11 

7.65 

8.19 

22 

5.i6 

5.53 

5.9i 

6.3o 

6.71 

7.!3 

1  1 

o.38 

0.41 

o.44 

56 

7.29 

7-84 

8.39 

24 

5.6o 

5.99 

6.4i 

6.83 

7.27 

7.72 

12 

o.46 

0.49 

0.52 

57 

7-46 

8.02 

8.58 

26 

6.o3 

6.45 

6.90 

7.35 

7.83 

8.3i 

i3 

o.54 

o.57 

0.61 

58 

7.  63 

8.20 

8.78 

28 

6.45 

6.91 

7.38 

7.87 

8.37 

8.89 

i4 

0.62 

0.66 

0.71 

59 

7-79 

8.38 

8.97 

3o 

6.86 

7.35 

7.85 

8.37 

8.91 

9-46 

i5 

0.71 

o.76 

0.81 

60 

7.96 

8.56 

9.!6 

82 

7.27 

7.78 

8.32 

8.87 

9-44 

IO.02 

16 

0.80 

0.86 

0.92 

61 

8.12 

8.73 

9-34 

34 

7.67 

8.21 

8.77 

9.35 

9.95 

IO.57 

J7 

0.90 

°-97 

i  .o4 

62 

8.27 

8.90 

9.52 

36 

8.06 

8.62 

9.22 

9.83 

10.46 

II  .  II 

18 

.01 

.08 

1.16 

63 

8.42 

9.06 

9.70 

38 

8.43 

9-o3 

9.65 

10.29 

10.95 

n.63 

'9 

.  12 

.20 

1.29 

64 

8.57 

9.22 

9.87 

4o 

8.80 

9.42 

10.07 

10.  74 

H.43 

12.  a 

20 

.24 

.33 

i  .42 

65 

8.72 

9.38 

io.o4 

42 

9.  16 

9.80 

10.48 

ii  .  i7 

11.89 

12.63 

21 

.36 

.46 

i.56 

66 

8.86 

9.53 

10.20 

44 

9.5i 

10.  17 

10.88 

1  1.  60 

12.34 

i3.u 

22 

.48 

.59 

i.7o 

67 

9.00 

9.67 

10.35 

46 

9-84 

10.54 

11.26 

12.  OI 

12.78 

i3.57 

23 

.61 

•73 

i.85 

68 

9-i3 

9.81 

10.  5o 

48 

10.  16 

10.88 

n.63 

I2.4o 

13.20 

l4«O2 

24 

.75 

.88 

2.OI 

69 

9.26 

9.95 

io.65 

5o 

io.48 

II  .22 

11.99 

12.  78 

i3.6o 

14.45 

25 

1.89 

2.03 

3.19 

7° 

9.  38 

10.  o9 

10.79 

62 

10.78 

ii.54 

12^33 

i3.i5 

13.99 

14.86 

26 

2.03 

2.18 

2.33 

71 

9.5o 

IO.2I 

10.93 

54 

11.07 

11.84 

12.65 

i3.5o 

i4.36 

i5.25 

27 

8.18 

2.34 

2.5o 

72 

9.61 

10.33 

ii.  06 

56 

H.34 

12.  l4 

12.97 

i3.83 

14.72 

i5.63 

28 

2.33 

2.5o 

2.68 

73 

9.71 

io.45 

ii.  18 

58 

ii  .60 

12  .42 

I3.27 

i4.i5 

i5.o5 

i5.99 

29 

2.48 

2.67 

2.86 

74 

9.82 

io.56 

ii  .3o 

60 

11.84 

12.68 

i3.55 

i4.44 

!5.37 

16.32 

3o 

2.64 

2.84 

3.o4 

75 

9.91 

10.66 

ii  .4i 

62 

12.07 

12.93 

i3.8i 

14.73 

i5.67 

16.64 

3i 

2.80 

3.oi 

3.23 

76 

I  O.OO 

io.76 

ii.  5i 

64 

12.29 

i3.i6 

14.06 

14.99 

15.95 

i6.94 

32 

2.97 

3.i9 

3.42 

77 

10.09 

10.85 

ii  .61 

66 

12.49 

i3.37 

14.29 

i5.24 

16.21 

I7.22 

33 

3.i4 

3.37 

3.6i 

78 

10.  17 

10.  93 

ii  .70 

68 

12.68 

i3.58 

i4.5o 

i5.46 

i6.45 

i7.47 

34 

3.3i 

3.55 

3.8o 

79 

10.24 

II  .01 

11.78 

70 

12.85 

i3.76 

14.70 

15.67 

16.67 

i7.7i 

35 

3.48 

3.74 

4.oo 

80 

10.  3i 

1  1.  08 

11.86 

72 

i3.oo 

13.92 

i4.88 

i5.86 

16.88 

I7.92 

36 

3.65 

3.93 

4.20 

81 

10.37 

ii.  i5 

11.93 

74 

i3.i4 

14.07 

i5.o4 

i6.o3 

I7.o6 

18.12 

37 

3.83 

4.12 

4.4i 

82 

10.42 

II.  21 

11.99 

76 

i3.27 

14.21 

i5.i8 

16.18 

I7.22 

18.29 

38 

4.01 

4.3i 

4.61 

83 

10.47 

II  .26 

I2.o5 

78 

i3.38 

14.32 

i5.3o 

i6.3i 

i7.36 

i8.43 

39 

4.19 

4.5o 

4.82 

84 

io.5i 

ii.  3i 

12.  10 

80 

i3.47 

14.42 

i5.4o 

16.42 

i7.47 

i8.56 

4o 

4.37 

4.70 

5.o3 

85 

10.55 

H.35 

12.  l4 

82 

i3.54 

i4.5o 

i5.49 

i6.5i 

i7.57 

18.66 

4i 

4.55 

4.90 

5.2486 

10.58 

n.38 

12.17 

84 

i3.6o 

i4.56 

i5.56 

i6.59 

i7.65 

18.74 

42 

4.74 

5.09 

5.4587 

1  0.60 

ii  .4o 

12.20 

86 

i3.64 

14.61 

i5.6o 

i6.64 

I7.70 

18.80 

43 

4.92 

5.29 

5.66,88 

10.62 

ii  .42 

12.22 

88 

i3.67 

i4.63 

i5.63 

16.67 

17.73 

i8.83 

44 

5.  ii 

5.49 

5.88.89 

10.63 

ii.43 

12.23 

90 

13.67  i4.63 

i5.64 

16.68 

17.74 

i8.85 

45 

5.2915.69 

6.09  90 

io.63 

ii.43 

12.23 

TABLE    XY. 


379 


Parallax  of  the  Sun  and  Planets  at  different  Altitudes. 


5 

Horizontal  Parallax. 

»; 
<i 

I" 

2"   3" 

4//  1  5/7 

6//  7/7 

8"  |  9" 

10" 

20"  30"  8".4  8".5  8".6 

8".7|8".8 

o 
0 

I  .O 

2.O 

3.o 

4.o 

5.o 

6.0 

7.0 

8.0 

9.0 

IO.O 

20.0 

3o.o 

8.4o 

8.5o 

8.60 

8.70 

8.80 

2 

I  .O 

2.O 

3.o 

4.o 

5.o 

6.0 

7.0 

8.0 

9.0 

IO.O 

20.0 

3o.o 

8.39 

8.49 

8.59 

8.69 

8.79 

2 

4 

1.0 

2.0 

3.o 

4.0 

5.o 

6.0 

7.0 

8.0 

9.0 

IO.O 

20.  o 

29.9 

8.38 

8.488.58 

8.68 

8.78 

i 

6 

1  .0 

2.0 

3.o 

4.0 

5.o 

6.0 

7.0 

8.0 

9.0 

9.9 

19.9 

29.8 

8.35 

8.45 

8.55 

8.65 

8.75 

6 

8 

1  .0 

2.0 

3.o 

4.o 

5.o 

5.9 

6.9 

7-9 

8.9 

9.9 

19.8 

29.7 

8.32 

8.42 

8.52 

8.62 

8.71 

8 

10 

1  .0 

2.0 

3.o 

3.9 

4.9 

5.9 

6.9 

7-9 

8.9 

9.8 

19.7 

29.5 

8.27 

8.37 

8.47 

8.57 

8.67 

10 

12 

I  .0 

2.0 

2.9 

3.9 

4.9 

5.9 

6.8 

7.8 

8.8 

9.8 

19.6 

29.3 

8.22 

8.3i 

8.4i 

8.5i 

8.61 

12 

i4 

1  .0 

I.9 

2.9 

3.9 

4-9 

5.8 

6.8 

7.8 

8.7 

9-7 

19.4 

29.  i 

8.i5 

8.25 

8.34 

8.44 

8.54 

1^ 

16 

1  .0 

I.9 

2.9 

3.8 

4.8 

5.8 

6.7 

7-7 

8.7 

9.6 

19.2 

28.8 

8.07 

8.17 

8.27 

8.36 

8.46 

16 

18 

I  .0 

I.9 

2.9 

3.8 

4.8 

5.7 

6.7 

7.6 

8.6 

9.5 

19.0 

28.5 

7-99 

8.08 

8.18 

8.27 

8.37 

18 

20 

0.9 

1.9 

2.8 

3.8 

4.7 

5.6 

6.6 

7.5 

8.5 

9-4 

18.8 

28.2 

7.89 

7-99 

8.08 

8.18 

8.27 

20 

22 

0.9 

1.9 

2.8 

3.7 

4.6 

5.6 

6.5 

7-4 

8.3 

9.3 

i8.5 

27.8 

7-79 

7.88 

7-97 

8.07 

8.16 

22 

240.9 
260.9 

1.8 

1.8 

2.73.7 

2.73.6 

4.6 
4.5 

5.5 
5.4 

6.4 
6.3 

7.3 
7.2 

8.2 

8.1 

9.1 
9.0 

i8.3 
18.0 

27.4 
27.0 

7.67 
7.55 

7-77 
7-64 

7.86 
7.73 

7.95 

7.82 

8.o4 
7.91 

"2.L 
26 

28 

0.9 

1.8 

2.6 

3.5 

4.4 

5.3 

6.2 

7-1 

7-9 

8.8 

17.7 

26.5 

7.42 

7.5i 

7.59 

7.68 

7-77 

28 

3o 

O  .  Q 

i-7 

2.6 

3.5 

4.3 

5.2 

6.1 

6.9 

7.8 

8.7 

!7.3 

26.0 

7.27 

7.36 

7.45 

7.53 

7.62 

3o 

32 

O.o 

i-7 

2.5 

3.4 

4.2 

5.  i 

5.9 

6!87.6 

8.5 

17.0 

25.4 

7.12 

7.21 

7.29 

7.  38  7.  46 

32 

34 

0.8 

J-7 

2.5 

3.3 

4.1  5.o 

5.8 

6.67.5 

8.3 

16.6 

24.9 

6.96 

7-o5 

7-i3 

7.2117.30 

34 

36 

0.8 

1.6 

2.4 

3.2 

4.04.9 

5.7 

6.57.3 

8.1 

16.2 

24.3 

6.80 

6.88 

6.96 

7.04:7.12 

36 

38 

0.8 

1.6 

2.4 

3.2 

3.9 

4-7 

5.5 

6.3 

7-i 

7-9 

i5.8 

23.6 

6.62 

6.70 

6.78 

6.866.93 

38 

to 

0.8 

i.5 

2.3 

3.i 

3.8 

4.6 

5.4 

6.  i 

6.9 

7-7 

i5.3 

23.0 

6.43 

6.5i 

6.59 

6.66 

6.74 

4o 

ia 

0.7 

i.5 

2.2 

3.o 

3.7 

4.5 

5.2 

5.96.7 

7-4 

i4.9 

22.3 

6.24 

6.32 

6.39 

6.47 

6.54 

42 

44 

0.7 

i.4 

2.2 

2.9 

3.6 

4.3 

5.o 

5.8 

6.5 

7.2 

i4.4 

21.6 

6.o4 

6.  ii 

6.  19 

6.26 

6.33 

44 

46 

0.7 

i.4 

2.  I 

2.8 

3.5 

4.2 

4.9 

5.6 

6.3 

6.9 

i3.9 

20.8 

5.84 

5.90 

5.97 

6.o4 

6.  ii 

46 

48 

0.7 

i.3 

2.0 

2.7 

3.3 

4.o 

4-7 

5.4 

6.0 

6.7 

i3.4 

20,  I 

5.62 

5.69 

5.75 

5.82 

5.89 

48 

5o 

0.6 

i.3 

1.9 

2.6 

3.2 

3.9 

4.5 

5.  i 

5.8 

6.4 

12.9 

19.3 

5.4o 

5.46 

5.53 

5.59 

5.66 

5o 

52 

0.6 

I  .2 

1.8 

2.5 

3.i 

3.7 

4.3 

4.95.5 

6.2 

12.3 

i8.5 

5.i7 

5.23 

5.29 

5.36 

5.42 

52 

54 

0.6 

I  .2 

1.8 

2.4 

2.9 

3.5 

4.i 

4-715.3 

5.9 

n.  8 

17.6 

4.94 

5.oo 

5.o5 

5.  ii 

5.i7 

54 

56 

0.6 

I  .  I 

1*7 

2.2 

2.8 

3.4 

3.9 

4.55.o 

5.6 

II  .2 

16.8 

4.70 

4.75 

4.8i 

4.86 

4.92 

56 

58 

o.5 

I  .  I 

1.6 

2.1 

2.6 

3.2 

3.7 

4.2 

4.8 

5.3 

10.6 

i5.9 

4.45 

4.5o 

4.56 

4.6i 

4.66 

58 

60 

o.5 

I  .0 

i.5 

2.0 

2.5 

3.o 

3.5 

4.0 

4.5 

5.o 

IO.O 

i5.o 

4.20 

4.25 

4.3o 

4.35 

4.4o 

60 

62 

o.5 

0.9 

i.4 

I.9 

2.3 

2.8 

3.3 

3.84.2 

4-7 

9.4 

i4.i 

3.94 

3.99 

4.o4 

4.o8 

4.i3 

62 

64 

0.4 

o  .9 

i.3 

1.8 

2.2 

2.6 

3.i 

3.53.9 

4.4 

8.8 

13.2 

3.68 

3.73 

3.77 

3.8i 

3.86 

64 

66 

o.4 

0.8 

I  .2 

1.6 

2.0 

2.4 

2.8 

3.23.7 

4.i 

8.1 

12.2 

3.42 

3.463.5o 

3.54 

3.58 

66 

68 

0.4 

0.7 

I  .  I 

i.5 

I.9 

2.2 

2.6 

3.o 

3.4 

3-7 

7,5 

II  .2 

3.i5 

3.i8 

3.22 

3.26 

3.3o 

68 

7° 

o.3 

0.7 

I  .O 

i.4 

•7 

2.1 

2.4 

2.7 

3.i 

3.4 

6.8 

io.3 

2.87 

2.91 

2.94 

2.98 

3.oi 

70 

72 

o.3 

0.6 

0.9 

I  .2 

,5i.9 

2.2 

2.5 

2.8 

3.i 

6.2 

9.3 

2.60 

2.63 

2.66 

2.69 

2.72 

72 

74 

o.3 

0.6 

0.8 

I  .  I 

•4II-7 

1.9 

2  .2 

2.5 

2.8 

5.5 

8.3 

2.32 

2.34 

2.3? 

2.40 

2.43 

74 

76 

O.2 

o.5 

0.7 

I  .O 

.2 

i.5 

J-7 

1-9 

2.2 

2.4 

4.8 

7.3 

2.03 

2.06 

2.08 

2.  10 

2.13 

76 

7  3 

0.2 

0.4 

0.6 

0.8 

.O 

I  .2 

i.5 

r-7 

I.9 

2.  I 

4.2 

6.2 

i.75 

1.77 

1.79 

1.81 

1.83 

78 

80 

O.2 

o.3 

o.5 

0.7 

o  9 

1.0 

1.2 

i.4 

1.6 

1-7 

3.5 

5.2 

1.46 

i.48 

1.49 

.5i 

i.53 

80 

82 

O.  I 

o.3 

o.4 

0.6 

0.7 

0.8 

I  .O 

I  .  I 

i.3 

i.4 

2.8 

4.2 

1.17 

1.18 

1  .20 

.21 

I  .22 

82 

84 

0.  I 

O.2 

o.3 

o.4 

o.5 

0.6 

0.7 

0.8 

0.9 

1  .0 

2.1 

3.i 

0.88 

0.89 

0.90 

0.91 

0.92 

84 

86 

O.  I 

O.  I 

O.2 

o.3 

o.3o.4 

0.5 

0.6 

0.6 

0.7 

i.4 

2.1 

o.59 

o.59 

0.60 

0.61 

0.61 

86 

88 

0.0 

O.  I 

O.I 

O.I 

O.2  O.2 

O.2 

o.3 

o.3 

o.3 

0.7 

I  .0 

0.29 

o.3o 

o.3o 

o.3o 

o.3i 

88 

90  o.o 

O.  I 

O.O 

o.o 

O.O  0.0 

O.O 

o.o 

o.o 

0.0 

O.O 

0.0 

o.oo 

0.00 

o.oo 

o.oo 

o.oo 

9° 

380  TABLE   XVI. — MOON'S  PARALLAX   IN  RIGHT  ASCENSION 


Hour 
Angle. 

Moon's  true  Declination,  0°. 

Moon's  true  Declination,  5°. 

Moon's  true  Declination,  10°. 

Hor 
53' 

zontal  Pars 
57' 

illax. 
61' 

Horizontal  Para 

53'     |     57' 

illax. 

61' 

Hori 
53' 

zontal  Para 

57' 

llax. 
61' 

Min. 

s. 

s. 

s. 

*. 

s. 

s. 

s. 

s. 

^ 

5 

3.47 

3.73 

3  .99 

3.48 

3.75 

4.01 

3.52 

3-79 

4.06 

10 

6.93 

7-46 

7-99 

6.96 

7-49 

8.02 

7.04 

7.58 

8.  ii 

i5 

io.3g 

11.19 

11.98 

io.43 

11.23 

12.  03 

10.55 

n.36 

12.17 

20 

13.85 

14.91 

i5.96 

13.90 

14.96 

i6.o3 

i4-o6 

i5.i4 

16.21 

25 

i7-3o 

18.62 

19.94 

17.36 

18.69 

20  .02 

17.57 

18.91 

20.26 

3o 

20.74 

22.32 

23.91 

20.82 

22.  4l 

24.00 

21  .06 

22.67 

24.28 

35 

24.17 

26.OI 

27.86 

24.26 

26.11 

27.97 

24.54 

26.42 

28.  3o 

4o 

27.59 

29.69 

3i.8o 

27.69 

29.81 

31.93 

28.02 

3o.i6 

32  .  3o 

45 

3o.99 

33.36 

35.  73 

3i.ii 

33.49 

35.87 

3i.47 

33.88 

36.29 

5o 

34.38 

37.01 

39.64 

34.  5i 

37.i5 

39.79 

34.92 

37.58 

40.26 

55 

37.75 

4o.64 

43.52 

37.9o 

40.79 

43.69 

38.34 

4l-27 

44.2O 

60 

4i  .  1  1 

44.25 

47.39 

41.27 

44-42 

47-57 

4i.75 

44.  94 

48.i3 

65 

44.44 

47.83 

5i.23 

44.6i 

48.02 

5i.43 

45.i3 

48.58 

62.  o3 

70 

47.75 

5i.4o 

55.  o5 

47-94 

5i  .60 

55.26 

48.  5o 

52.20 

66.91 

75 

5i.o4 

54.94 

58.84 

5i.24 

55.i5 

59.o7 

5i.84 

55.80 

69.76 

80 

54.  3o 

58.45 

62.60 

54.  5i 

58.68 

62.84 

55.  i5 

59.36 

63.58 

85 

57.54 

6i.93 

66.33 

57.76 

62.  17 

66.  59 

58.44 

62.  9o 

67.37 

9° 

60.75 

65.  39 

70.  o3 

60.98 

65.64 

70.  3o 

61.70 

66.  4i 

7I  .  12 

95 

63.  93 

68.  81 

73.69 

64.17 

69.07 

73.98 

64.92 

69.88 

74.84 

IOO 

67.08 

72  .20 

77.32 

67.33 

72.47 

77.62 

68.12 

73.32 

78.53 

io5 

70.19 

75.55 

80.91 

70.46 

75.84 

81.22 

71.28 

76.72 

82.17 

no 

73.27 

78.86 

84.46 

73.55 

79.16 

84.78 

74.  4i 

80.  o9 

86.77 

ii5 

76.3i 

82.14 

87.97 

76.61 

82.45 

88.  3o 

77.  5o 

83.42 

89.34 

I2O 

79.32 

85.  37 

91.43 

79.63 

85.  70 

91-78 

80.  56 

86.70 

92.86 

125 

82.28 

88.56 

94.85 

82.60 

88.90 

95.21 

83.57 

89.  95 

96.33 

i3o 

85.2i 

91.71 

98.22 

85.54 

92.07 

98.60 

86.54 

93.14 

99.76 

1.35 

88.10 

94.82 

101  .54 

88.44 

96.18 

101  .93 

89.47 

96.29 

I03.I2 

i4o 

90.94 

97.88 

104.82 

91.29 

98.25 

105  .22 

92.36 

99.40 

106.45 

i45 

93.74 

ioo.89 

*io8.o4 

94.  10 

101  .27 

108.45 

95.20 

102  .46 

109.72 

i5o 

96.49 

io3.85 

III  .21 

96.86 

104.24 

in.  63 

97-99 

io5.46 

112.94 

i55 

99.20 

i  06.  76 

II4.32 

99.58 

107.17 

i  14.76 

100.74 

108.42 

116.  10 

160 

101.85 

109.61 

117.38 

102.24 

no.o3 

117.  83 

io3.44 

III.  32 

119.21 

170 

107.01 

iiS.iG 

123.32 

107.43 

ii5.6i 

123.  80 

108.68 

116.96 

126.24 

180 

111.97 

I20.49 

129.02 

II2.40 

120.96 

129.52 

113.71 

122.37 

i3i.o3 

190 

116.  70 

125.58 

134.47 

II7.l5 

126.07 

i  34.  99 

n8.52 

127.54 

i36.57 

200 

121  .21 

i3o.43 

139.66 

121  .67 

180.93 

i4o.2o 

123.09 

i32.46 

i4i.83 

210 

i*2  5.  48 

i  35.  02 

i44.57 

125.96 

i35.54 

i45.i3 

i27.43 

137.12 

146.82 

220 

129.  5i 

i39.35 

149.20 

i3o.oi 

139.89 

149.78 

i3i.52 

i4i  .52 

161.62 

230 

133.29 

i43.4i 

i53.54 

i33.8o 

143.96 

:54.i3 

135.36 

i45.64 

i  55.  93 

240 

i36.8o 

147-19 

i57.59 

137.33 

147.76 

168.19 

i38.93 

149.48 

i6o.o3 

25o 

i4o.o6 

i5o.69 

161.32 

i4o.6o 

i5i  .27 

161.94 

142.23 

i53.o3 

163.83 

260 

i43.o4 

i53.89 

i64.75 

i43.59 

i54.48 

165.38 

145.26 

i56.28 

167.30 

270 

i45.75 

i56.8o 

167.85 

i46.3i 

157.40 

i68.5o 

i48.oo 

169.23 

170.46 

280 

148.17 

i59.4o 

170.63 

i48.74 

160.01 

171  .29 

160.47 

161.87 

173.28 

290 

i5o.3i 

161  .  70 

173.09 

i5o.89 

162.32 

i73.75 

i52.64 

164.20 

176.  77 

3oo 

i52.i6 

163.68 

175  .20 

152.75 

i64.3i 

i75.87 

i54.52 

166.22 

I77.92 

3io 

i53.72 

165.35 

176.99 

i54.3i 

165.99 

177.66 

i56.  10 

167.91 

I79.72 

320 

154.99 

166.71 

178.43 

i55.58 

167.34 

179.11 

i57.38 

169.28 

181  .19 

33o 

i55.95 

i67.74 

179.53 

i56.55 

168.38 

180.22 

i58.36 

170.33 

182.  3o 

34o 

i56.62 

i68.45 

180.29 

157.22 

169.10 

180.98 

i59.o4 

171  .o5 

183.07 

35o 

i56.99 

168.85 

180.70 

i57.59 

169.49 

i8i.39 

i59.4a 

i7i.45 

i83.49 

36o 

157.06 

i68.92 

180.77 

i57.66 

169.  56 

181.46 

169.49 

171  .62 

i83.56 

370 

i57.43 

169.31 

181.18 

169.26 

171  .26 

i83.27 

38o 

i58.72 

170.68 

182.64 

FOR   CAMBRIDGE   OBSERVATORY,  LAT.  42°  22'  48".6.    381 


Hour 
Angle. 

Moon's  true  Declination,  15°. 

Moon's  true  Declination,  20°. 

Moon's  true  Declination,  25°. 

Hor 

53' 

zontal  Parallax. 
57'            61' 

Horizontal  Parallax. 

53'         57'          61' 

Horizontal  Para 

53'     |     57' 

llax. 

61' 

Min. 

5 

10 

i5 

20 
25 

3o 

3% 

7.17 
10.76 
i4.34 
17.91 

21.48 

*. 
3.86 
7.73 
11.59 

i5.44 
19.28 

23.12 

4*'i4 
8.28 

12.  4l 

16.54 
20.65 
24.76 

3*.'  6  9 
7.  38 
11.07 
l4.74 

18.42 
22.08 

3%7 

7.94 
11.91 
16.87 
19.83 
23.77 

4*26 
8.5i 

12  .  76 
17.00 
21  .  24 

25.46 

3*.'  8  3 
7.66 
11.48 
i5.3o 
19.11 

22  .91 

4.12 
8.24 
12.36 
16.47 
20.67 
24.66 

4*4i 

8.83 

13.24 

17.64 

22.  03 

26.4l 

35 
4o 
45 
5o 
55 
60 

25.  o3 

28.57 

32.10 

35.61 
39.10 

42.57 

26.94 

3o.75 
34.55 
38.33 
42.09 

45.83 

28.86 
32.  94 

37.01 
4i.o5 
45.o8 
49.08 

25.  74 
29.38 
33.oo 
36.  61 

40.20 
43.78 

27.71 
3i.62 
35.53 
39.4i 
43.28 
47.12 

29.68 
33.87 
38.  o5 
4s.  2  1 
46.36 
5o.47 

26.70 

3o.47 
34.23 
37.98 
4i  .70 
45.  4i 

28.74 

32.  80 

36.85 
4o.88 
44.89 
48.88 

3o.78 
35.i4 
39.47 
43.79 
48.09 
52.36 

65 
70 

75 
80 
85 
9° 

46.o3 
49.45 
52.86 
56.24 
59.59 
62  .91 

49-54 
53.23 
56.90 
6o.54 
64.i4 
67.72 

53.o6 
57.02 
60.94 

64.84 

68.70 
72.53 

47-32 

5o.85 
54.35 
57.83 
61.28 
64.69 

50.94 
54.74 
58.  5i 
62.25 
65.  96 
69.63 

54.  57 
58.63 
62.67 
66.67 
70.65 
74.58 

49.09 
52.  75 
56.38 
59.98 
63.56 
67.  10 

52.84 
56.  78 
60.69 
64.  57 
68.42 
72.23 

56.  60 
60.82 
65.01 
69.16 
73.28 
77.37 

95 

100 

io5 
no 
ii5 

120 

b'6.  21 

69.47 
72.69 
75.88 
79.  o3 

82.14 

71  .26 
74  .  77 
78.24 
81.67 
85.o6 
88.  4i 

76.32 
80.08 

83.  80 

87.47 
91.10 
94.69 

68.08 
71.43 
74.74 
78.02 
81.26 
84.46 

73.28 
76.88 
8o.45 
83.  98 

87.47 
90.91 

78.48 
82.34 
86.17 
89.95 
93.68 
97.37 

70.61 
74.09 
77.53 
80.93 

84.29 
87.61 

76.01 
79.76 
83.45 
87.11 
90.73 
94.  3o 

8i.4i 
86.42 
89.  38 
93.3o 
97-  1? 

101  .00 

125 

i3o 
1  35 

i4o 
i45 
i5o 

85.22 

88.25 
91  .24 
94.18 

97.08 
99.93 

91.72 
94.98 
98.20 
101.36 
io4.48 
107.55 

98.23 
101  .72 
io5.  17 
io8.55 
in  .89 
n5.  17 

87.62 
90.74 
93.8i 
96.84 
99.81 

102  .74 

94.3i 
97.67 
100.97 

104.23 

107.43 

no.58 

101  .OI 

io4-6o 
108.14 
111.62 
ii5.o5 

118.42 

90.88 
94.11 
97.  3o 
100.44 
io3.53 
106.57 

97.82 
101  .3o 
104.73 
1  08.  n 
in.  43 
114.7° 

io4.77 
108.60 

112.  I7 
Il5.78 
119.34 

122.84 

l55 

1  60 
170 
180 
190 

200 

102  .  73 

io5.48 
110.82 
ii5.95 

120.85 
1^5.52 

no.56 
ii3.52 
119.27 
124.78 
i3o.o5 
i35.o7 

118.40 
121  .56 
127.72 
133.62 
139.26 
i44.63 

io5.62 

108.45 

i  i  3  .  94 
119.21 

124.25 
129.05 

n3.68 
116.72 

122  .63 

128.  3o 
i33.72 
i38.87 

121  .74 
124.99 

i3i.32 
i37.39 
143.19 

148.70 

109.55 
ii2.48 
118.18 
123.65 
128.87 
133.84 

117.91 

121  .06 
I27.I9 

i33.o7 
i38.69 
i44.o3 

126.27 
129.66 
i36.2i 
142.60 
i48.5i 
i54.23 

2IO 

220 
230 
24o 

25o 

260 

i^y.94 
i34.  i  i 
i38.o2 
i4i.66 
i45.o2 
i  48.  i  i 

139.82 
i44-3i 
i48.5i 

152.42 

i56.o3 

i59.35 

149.71 
i54.5i 
159.00 
i63.i8 
167.05 
170.59 

i33.59 
137.88 
i4i  .90 
i45.64 
149.  10 
162.27 

143.76 
148.36 
i  52.  68 
156.70 
160.42 
i63.82 

153.93 

i58.85 

i63.47 
167.77 

I7L74 

i75.38 

138.55 
i43.oo 
147.16 
i5i.o3 
i54.62 
157.90 

i49.  10 
i53.87 
i58.35 
162.61 
i66.36 
i69.89 

i59.65 
164.76 
i69.54 
i73.99 
178.11 
181.88 

270 
280 
290 

3oo 
3io 

320 

i  50.91 
i53.42 
i55.63 
i57.55 
159.16 
160.46 

i62.35 
i65.o5 
167.42 
169.48 

171  .20 
172.60 

173.80 
176.68 
179.22 
181.41 
i83.25 
i84.74 

i  55.i4 

i57.72 
159.99 
161  .96 
i63.6i 
164.95 

166.91 
169.67 
172.  n 
174.22 

175.99 

177.43 

i78.68 
i8i.63 
184.24 
186.49 
188.38 
189.91 

160.89 
163.55 
165.91 
167.94 
169.65 
171.04 

I73.o9 
i75.95 
i78.48 
180.66 
182.  49 
i83.98 

i85.3o 
188.36 
191  .06 
193.38 
196.34 

I96.  92 

33o 
34o 
35o 
36o 

370 

38o 

161  .40 
162.  i5 
i62.53 
162  .60 
162.37 
161.82 

173.67 
174.40 
174.80 
174.87 
!74.6l 
174.02 

185.87 
i86.65 
187.08 
187.14 
i86.85 
186.21 

165.98 
166.68 
167.07 
167.14 
166.90 
i66.33 

178.52 
179.28 
179.69 

I79-75 
179.48 
178.87 

191.07 
191.87 
192.30 
192.37 
192.07 
i9i  .4o 

172.  10 
172.  83 
i73.23 
i73.3o 
173.04 
172.45 

i85.ii 
i85.89 
i86.3i 
186.38 
186.09 
i85.45 

198.12 
198.96 
199.39 
199.45 
I99.I4 
198.44 

390 
4oo 
4io 
420 

160.97 
iSg.Si 

173.09 

171.  84 

i85.2i 
18.3.87 

i65.45 
164.26 
162.75 
160.94 

177.91 
176.62 
175.00 
173.04 

190.37 
188.99 
187.24 
i85.i4 

171.53 
170.29 
168.73 
i66.85 

i84.45 
i  83.  i  i 
i8i.43 
1-79.40 

197.37 
196.93 
194.12 
191  .94 

382     TABLE    XVI. — MOON'S   PARALLAX   IN    DECLINATION 


Moon's  true  Declination,  0°. 

Moon's  true  Declination,  5°  N. 

Moon's  true  Declination,  10°  N. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Hour 

Angle. 

53' 

57' 

61' 

53' 

57' 

61' 

53'             57' 

61' 

m. 

" 

// 

O 

2160.  i 

2326.2 

2490.6 

1945.9 

2094.7 

2243.8 

I7I6.3 

1847.7 

1979.8 

20 

2160.0 

2325.1 

2490.4 

1946.6 

2096.6 

2244.6 

1717.8 

1849.8 

1981  .  i 

4o 

2159.8 

2324.7 

2490.0 

1948.8 

2097.8 

2247.0 

1722.4 

1864.2 

1986.3 

60 

2169.3 

2324.2 

2489.3 

1962.3 

2IOI.5 

2261  .0 

1729.9 

1862.3 

1994.9 

80 

2i58.6 

2323.4 

2488.4 

1967.2 

2106.  8 

2266.6 

1740.4 

i873.5 

2006.9 

100 

2167.8 

2322  .4 

2487.3 

1963.6 

2ii3.5 

2263.7 

i753.7 

1887.8 

2O22  .2 

120 

2166.  8 

2321.3 

2486.0 

1971.0 

2I2I.5 

2272.2 

1769.7 

1906.0 

2o4o.5 

i4o 

2166.  6 

2319.9 

2484.4 

1979.7 

2i3o.8 

2282.1 

1788.3 

1926.0 

2061.8 

160 

2i54.3 

23i8.3 

2482.6 

1989.6 

2141.4 

2293.3 

1809.4 

1947.6 

2086.0 

180 

2162.8 

23i6.6 

2480.7 

2OOO.6 

2i53.i 

2306.7 

1882.8 

1972.6 

2II2.7 

200 

2l5l  .2 

23i4.8 

2478.6 

2OI2.5 

2166.8 

2319.2 

1868.2 

1999.9 

2l4l.8 

220 

2149.4 

2312.8 

2476.3 

2026.3 

2179.4 

2333.7 

1886.4 

2029.  i 

2178.0 

24o 

2147.6 

2310.7 

2473.8 

2038.9 

2193.9 

2849.1 

1914.4 

2060  .  i 

2206  .  I 

260 

2146.  7 

23o8.4 

2471.3 

2o53.i 

2209.  i 

2365.2 

1944.7 

2092.6 

2240.8 

280 

2i43.7 

23o6.i 

2468.6 

2067.9 

2224.9 

2881.9 

1976.2 

2126.4 

2276.8 

3oo 

2141.7 

23o3.8 

2466.9 

2083.2 

224l  .  I 

2899.  i 

2008.7 

2l6l  .2 

2818.9 

820 

2139.6 

23oi.3 

2463.1 

2098.7 

2267.7 

2416.  7 

2041  .8 

2196.  7 

2861.7 

34o 

2i37.5 

2298.9 

246o.3 

2ii4.5 

2274.4 

2434.4 

2076.3 

2282.6 

2889.9 

36o 

2i35.3 

2296.4 

2467.6 

2i3o.3 

2291  .3 

2462.8 

2109.0 

2268.7 

2428.4 

3  80 

2142.6 

2804.6 

2466.7 

MOON'S    PARALLAX   IN   DECLINATION 

Moon's  true  Declination,  0°. 

Moon's  true  Declination,  5°  S. 

Moon's  true  Declination,  10°  S. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Hour 
Angle. 

53' 

57' 

61' 

53' 

57' 

61' 

53' 

57' 

61' 

m. 

" 

" 

>• 

" 

" 

" 

// 

// 

0 

2160.  i 

2325.2 

2490.6 

2357.3 

2637.3 

2717.4 

2536.1 

2-729.4 

2928.0 

2O 

2160.0 

2325.1 

2490.4 

2356.4 

2536.3 

2716.4 

2534.4 

2727.6 

2921  .0 

4o 

2169.8 

2324.7 

2490.0 

2353.8 

2533.4 

2718.2 

2629.8 

2722.  I 

2916.  I 

60 

2169.3 

2324.2 

2489.3 

2349-3 

2628.6 

2708.0 

2620.9 

2712.9 

2906.3 

80 

2168.  6 

2323.4 

2488.4 

2343.1 

2621  .7 

2700.7 

2609.  i 

2700.  2 

2891.6 

IOO 

2167.8 

2322.4 

2487.3 

2335.2 

25i3.i 

2691  .4 

2494.2 

2684.1 

2874.3 

I2O 

2166.8 

2321.8 

2486.0 

2325.6 

2602.8 

2680.2 

2476.3 

2664.  7 

2868.4 

i4o 

2i55.6 

2319.9 

2484.4 

23i4.5 

2490.8 

2667.2 

2455.5 

2642.2 

2829.  I 

160 

2i54.3 

23i8.3 

2482.6 

23O2.  I 

2477.2 

2662.6 

2481.9 

2616.7 

2801  .7 

180 

2162.8 

23i6.6 

2480.7 

2288.2 

2462.  i 

2686.2 

24o5.8 

2688.6 

2771.3 

200 

2l5l  .2 

23i4.8 

2478.6 

2273.  I 

2445.8 

2618.6 

2877.4 

2667.7 

2788.2 

220 

2149.4 

2312.8 

2476.3 

2266.9 

2428.2 

2699.6 

2846.9 

2624.7 

2702.7 

24o 

2147.6 

23io.  7 

2473.8 

2289.7 

2409.5 

2679.4 

23i4.5 

2489.7 

2666.0 

260 

2145.7 

2808.4 

2471.3 

2221  .7 

2389.9 

2668.8 

2280.6 

2453.0 

2626.5 

280 

2143.7 

23o6.i 

2468.6 

22O3.0 

2869.6 

2536.4 

2245.3 

2414.9 

2684.5 

3oo 

2141.7 

23o3.8 

2466.9 

2l83.7 

2348.7 

2618.8 

2209.0 

2375.7 

2642.4 

320 

2189.6 

23oi  .3 

2463.1 

2l64.0 

2827.4 

2490.8 

2171.9 

2335.7 

2^99-3 

34o 

2137.6 

2298.9 

246o.3 

2i44«  i 

23o5.8 

2467.6 

36o 

2i35.3 

2296.4 

2467.5 

FOR   CAMBRIDGE   OBSERVATORY,   LAT.  42°  22'  48". 6.    383 


Moon's  true  Declination,  15°  N. 

Moon's  true  Declination,  20°  N. 

Moon's  true  Declination,  25°  N. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Hour 
Angle. 

53' 

57' 

61' 

53' 

57' 

61' 

53'          57'          61' 

m. 

0 

1473.2 

i586.0 

1699.0 

1218.3 

i3u.7 

i4o5.2 

c;53.8 

1026.1,    1100.2 

2O 

1475.5 

i588.5 

1701.7 

1221.4 

i3i5.o 

i4o8.8 

957.6 

io3i.i 

no4.7 

4o 

1482.4 

i595.9 

1709.  7 

1280.7 

i325.o 

i4i9.5 

969.2 

io43.5 

1118.0 

60 

1493.9 

1608.3 

1722.9 

1246.0 

i34i.5 

l437.2 

988.3 

1064.1 

n4o.i 

80 

i  Sog.  8 

1625.4 

i74i.3 

1267.4 

1364.5 

i46i.8 

!0l4.8 

1092.7 

1170.7 

IOO 

i53o.i 

i647.2 

I764.6 

1294.5 

i393.6 

i493.o 

io48.6 

1129.0 

1209.6 

I2O 

i554.5 

i673.5 

i792.7 

1327.1 

1428.8 

i53o.6 

1089.2 

1172.8 

1256.5 

i4o 

1582.9 

i  7o4  .  o 

i825.3 

i365.i 

i469.6 

i574.3 

n36.fi 

1223.6 

1810.9 

160 

i6i5.o 

i738.5 

1862.2 

i4o8.o 

i5i5.8 

1623.8 

1189.9 

1281.1 

1872.5 

180 

i65o.6 

i776.7 

1903.1 

1455.6 

i566.9 

1678.5 

1249.1 

1344.8 

1440.7 

200 

1689.3 

i8i8.3 

1947.6 

i5o7.3 

1622.6 

I738.i 

1818.6 

i4i4-2 

i5i5.o 

220 

1730.9 

i863.o 

1995.3 

1562.9 

1682.3 

1802.0 

1882.7 

1488.6 

i594.7 

240 

i775.o 

1910.4 

2045.9 

1621  .9 

i745.7 

1869.8 

i456.i 

i567.5 

i679.2 

260 

1821.3 

1960.0 

2098.9 

i683.7 

l8l2.  2 

1940.9 

i533.i 

i65o.3 

1767.8 

280 

1869.3 

20II.6 

2i54.i 

1747-9 

1881.2 

2014.7 

1618.  i 

i736.3 

1859.8 

3oo 

1918.8 

2064.  7 

22IO.8 

1814.1 

i952.3 

2090.  6 

i695.4 

1824.8 

1954.4 

320 

1969.3 

2118.9 

2268.6 

i  881.6 

2024.8 

2168.1 

1779.5 

1915.2 

2o5i  .0 

34o 

2020.4 

2I73.7 

2327.  I 

1949.9 

2098.2 

2246.5 

1864.6 

2006.6 

2148.8 

36o 

2071.7 

2228.8 

2385.  9 

2018.6 

2I7I.9 

2325.3 

1950.  i 

2098.4 

2246.9 

38o 

2122.9 

2283.7 

2444.5 

2o87.  i 

2245.4 

24o3.8 

2035.3 

2190.0 

2844.8 

4oo 

2i73.6 

2338.0 

25O2.5 

2i54.8 

23i8.i 

2481.4 

2119.7 

2280.6 

2441.6 

420 

2221.4 

2389.5 

2557.6 

2202  .  5 

2869.5 

2536.5 

44o 

2283.2 

2456.1 

2629.0 

FOR   CAMBRIDGE   OBSERVATORY,   LAT.  42°  22'  48".6. 

Moon's  true  Declination,  15°  S. 

Moon's  true  Declination,  20°  S. 

Moon's  true  Declination,  25°  S. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Horizontal  Parallax. 

Hour 
Angle. 

53' 

57' 

61' 

53' 

57' 

61' 

53' 

57' 

61' 

m. 

" 

" 

" 

" 

" 

" 

" 

" 

" 

0 

2694.9 

2900.  i 

3io5.6 

2832.8 

3o48.2 

3263.8 

2948.6 

3i72.5 

3396.6 

2O 

2692.5 

2897.5 

3l02.7 

2829.5 

3o44.7 

3260.1 

2944.6 

8168.2 

8892.0 

4o 

2685.o 

2889.4 

3094.0 

2819.8 

3o34.2 

8248.8 

2982.7 

3i55.4 

8878.2 

60 

2672.7 

2876.1 

30-79.7 

28o3.8 

3oi6.9 

3280.2 

2918.0 

3i34.i 

3355.3 

80 

2655.6 

2867.6 

8059.9 

278  i  v4 

2992.7 

8204.2 

2885.6 

3io4.5 

3323.6 

IOO 

2633.9 

2834.2 

3o34.6 

2753.o 

2962  .1 

8171.8 

285o.8 

3o67.o 

3283.3 

120 

26o7.7 

2805.9 

3oo4.2 

27i8.8 

2925.  I 

3i3i.6 

2808.9 

8021  .7 

8234.7 

i4o 

2577.3 

2773.0 

2968.9 

2679.  i 

2882.2 

3o85.5 

2760.  i 

2969.  I 

3i78.2 

160 

2542.9 

2735.9 

2928.9 

2634.1 

2883.  7 

3o33.4 

2705.0 

2909-7 

3u4.4 

180 

25o4.8 

2694.7 

2884.7 

2584.3 

2780.0 

2975.7 

2648.9 

2843.8 

8043.7 

200 

2463.3 

2649.8 

2836.5 

253o.i 

2721  .5 

2912.9 

2577.4 

2772.0 

2966.7 

220 

24i8.7 

2601  .7 

2784.8 

2471.9 

2658.7 

2845.5 

25o6.o 

2695.  I 

2884.1 

24o 

237i.5 

255o.7 

2780.0 

2692.  i 

2774.0 

2430.3 

26i3.4 

2796.5 

260 

2321  .9 

2497.2 

2672.5 

2  345  .*4 

2522.2 

2699.0 

235o.9 

2527  .8 

2704.7 

280 

2270.4 

244i.6 

2612.9 

2277.I 

2449.6 

2621  .  i 

3oo 

2217.4 

2384.5 

255i.5 

384 


TABLE   XVII. 

Parallactic  Angles  for  the  Latitude  of  Washington  Observatory. 


Hour  Angle. 

Dec. 

100 

20o 

30o   |    4Qo 

50o 

60o 

70o 

80o       90o 

lOOo      1100 

o 

0 

o 

0 

0 

0 

0 

0 

0 

o 

0 

0 

29  N. 

37.8 

54.4 

60.7 

62.9 

63.i 

62.1 

60.3 

58.o 

55.o 

5i.4 

47.3 

28 

35.3 

52.1 

59.0 

61.6 

62.  i 

61.4 

59.8 

57.6 

54.7 

5i.3 

47«3 

27 

33.i 

5o.o 

57.3 

60.3 

61.1 

60.7 

Sg.S 

57.2 

54.5 

5l.2 

47.2 

26 

3i.o 

48.o 

55.7 

59.  1 

60.2 

60.0 

58.7 

56.8 

54.2 

5i  .0 

47.2 

25 

29.2 

46.i 

54.2 

58.o 

59.3 

59.3 

58.2 

56.5 

54.o 

50.9 

47.2 

24 

27.6 

44-4 

52.8 

56.8 

58.4 

58.6 

57.7 

56.i 

53.8 

5o.8 

47.2 

23 

26.2 

42.7 

5i.4 

55.7 

57.6 

57.9 

57.3 

55.8 

53.6 

5o.7 

47.2 

22 

24.9 

4i  .2 

5o.i 

54.7 

56.8 

57.3 

56.8 

55.5 

53.4 

5o.6 

47.2 

21 

23.  7 

39.8 

48.8 

53.6 

55.9 

56.  7 

56.3 

55.2 

53.2 

5  0.6 

2O 

22.6 

38.4 

47.6 

52.6 

55.2 

56.i 

55.9 

54.8 

53.0 

5o.5 

19 

21.7 

37.! 

46.4 

5i.7 

54.4 

55.5 

55.5 

54.6 

52.8 

5o.4 

18 

20.8 

36.o 

45.3 

5o«7 

53.7 

54.9 

55.i 

54.3 

52.7 

5o.4 

17 

19.9 

34.8 

44.3 

49-8 

52.9 

54-4 

54-7 

54.0 

52.5 

5o.3 

16 

19.2 

33.8 

43.2 

49.0 

52.2 

53.9 

54.3 

53.7 

52.4 

5o.3 

i5 

i8.5 

32.8 

42.3 

48.2 

5i.6 

53.3 

53.9 

53.5 

52.3 

5o.3 

i4 

17.8 

31.9 

4i.4 

47.4 

50.9 

52.8 

53.5 

53.3 

52.1 

5o.2 

i3 

17.2 

3i.o 

4o.5 

46.6 

5o.3 

52.3 

53.2 

53.o 

52.  O 

5o.2 

12 

16.7 

30.2 

39.6 

45.8 

49.7 

5i.9 

52.8 

52.8 

5i.9 

5o.2 

II 

16.2 

29.4 

38.8 

45.i 

49.1 

5i.4 

52.5 

52.6 

ftiij 

IO 

i5.7 

28.7 

38.o 

44.4 

48.5 

5i.o 

52.2 

62.4 

61.7 

9 

15.2 

28.0 

37.3 

43.7 

48.o 

5o.5 

5i.9 

52.2 

5i.6 

8 

i4.8 

27.3 

36.6 

43.i 

47-4 

5o.i 

5i.6 

52.  O 

5i.6 

7 

i4.4 

26.7 

35.9 

42.5 

46.  9 

49.7 

5i.3 

5i.8 

5i.5 

6 

14.0 

26.  1 

35.3 

41.9 

46.4 

49.3 

5i.o 

5i.7 

5i.4 

5 

i3.7 

25.5 

34-7 

4i.3 

45.9 

48.  9 

50.7 

5i.5 

5i.4 

4 

i3.4 

25.0 

34.i 

40.7 

45.4 

48.6 

5o.5 

5i.4 

5i.4 

3 

i3.i 

24.5 

33.5 

40.2 

45.o 

48.2 

5o.2 

5l.2 

5i.3 

2 

12.8 

24.0 

33.o 

39.7 

44.5 

47-9 

5o.o 

5i.i 

5i.3 

i  N. 

12.5 

23.5 

32.5 

39.2 

44.i 

47.5 

49.8 

5i.o 

5i.3 

o 

12.2 

23.1 

32.0 

38.7 

43.7 

47.2 

49.5 

50.9 

5i.3 

i  S. 

12.0 

22.7 

3i.5 

38.3 

43.3 

46.  9 

49.3 

5o.8 

2 

II.7 

22.3 

3i.o 

37.8 

42.9 

46.6 

49.1 

5o.7 

3 

II.  5 

21.9 

3o.6 

37.4 

42.6 

46.3 

49.0 

5o.6 

4 

ii.  3 

21  .6 

30.2 

37.0 

42.2 

46.i 

48.8 

5o.5 

5 

II.  I 

21  .2 

29.8 

36.6 

41.9 

.45.8 

48.6 

5o.5 

6 

10.9 

20.9 

29.4 

36.2 

4i.6 

45.6 

48.5 

5o.4 

7 

10.7 

2O.  6 

29.0 

35.9 

4l.2 

45.3 

48.3 

5o.3 

8 

10.6 

20.3 

28.7 

35.5 

40.9 

45.i 

48.2 

5o.3 

9 

10.4 

20.0 

28.3 

35.2 

40.7 

44-9 

48.o 

5o.3 

10 

10.2 

19.8 

28.0 

34-9 

4o.4 

44.  i 

47-9 

60.2 

ii 

IO.  I 

I9.5 

27.7 

34.6 

4o.  i 

44.5 

47-8 

5o.2 

12 

IO.O 

I9.3 

27.4 

34.3 

39.9 

44.3 

47-7 

5o.2 

i3 

9.8 

19.0 

27.1 

34.o 

39.6 

44.i 

47.6 

5o.2 

i4 

9-7 

18.8 

26.9 

33.7 

39.4 

43.9 

47-5 

i5 

9.6 

18.6 

26.6 

33.5 

39.2 

43.8 

47-5 

16 

9.5 

18.4 

26.4 

33.2 

39.0 

43.6 

47-4 

i? 

9-4 

18.2 

26.  i 

33.o 

38.8 

43.5 

47-3 

18 

9.3 

18.0 

25.9 

32.8 

38.6 

43.4 

47-3 

19 

9.2 

17.8 

25.7 

32.5 

38.4 

43.3 

47.2 

20 

9.1 

17.7 

25.5 

32.3 

38.2 

43.i 

47.2 

21 

9.0 

i7.5 

25.3 

32.2 

38.i 

43.o 

47.2 

22 

8.9 

i7.3 

25.1 

32.0 

37.9 

43.o 

47.2 

23 

8.8 

17.2 

24.9 

3i.8 

37.8 

42.9 

47-2 

24 

8.7 

17.1 

24.8 

3i.6 

37.6 

42.8 

47.2 

25 

8.6 

16.9 

24.6 

3i.5 

37.5 

42.  7 

26 

8.6 

16.8 

24-4 

3i.3 

37.4 

42.  7 

27 

8.5 

16.7 

24.3 

3l.2 

37.3 

42.6 

28 

8.4 

16.6 

24.2 

3i.i 

37.2 

42.6 

29  S. 

8.4 

i6.5 

24.0 

So.g 

37.i 

42.6 

TABLE    XVIII.  385 

Correction  to  be  added  to  the  Moon's  Declination  in  computing  an  Occultation  or  Eclipse. 


'"'  '  "r      Difference  of  Right  Ascension. 

Dec. 

5' 

10' 

15' 

20' 

25' 

30' 

35' 

40' 

45' 

50'  55' 

60' 

65' 

70'    75' 

80' 

85' 

9(y 

0         / 

O        O 

0.0 

0.0 

o.o 

o.o 

o.o 

0.0 

o.o 

o.o 

o.o 

o.o 

o.o 

0.0 

o.o 

o.o 

o.o 

o.o 

0.0 

o.o 

o  3o 

0.0 

o.o 

o.o 

0.0 

o.o 

0.  I 

0.  I 

0.  I 

O.2 

0.2 

O.2 

o.3 

o.3 

0.4 

0.4 

o.5 

o.5 

0.6 

I        0 

0.0 

o.o 

o.o 

0.  I 

0.  I 

O.I 

O.2 

O.2 

o.3 

0.4 

o.5 

o.5 

0.6 

0.7 

0.9 

I.O 

i  .  i 

1.2 

i    3o 

o.o 

0.0 

O.I 

O.I 

0.  I 

0.2 

0.3 

0.4 

o.5 

0.6 

0.7 

0.8 

I  .0 

i.i 

i.3 

i.5 

x-7 

1.9 

2       0 

0.0 

o.o 

0. 

O.  I 

O.2 

o.3 

0.4 

o.5 

0.6 

0.8 

0.9 

i  .  i 

i.3 

i.5 

i-7 

1.9 

2.2 

2.5 

2    30 

o.o 

o.o 

o. 

0.2 

O.2 

o.3 

o.5 

0.6 

0.8 

.0 

1.2 

i.4 

1.6 

1.9 

2.  I 

2.4 

2.8 

3.i 

3     o 

o.o 

0.0 

o. 

O.2 

o.3 

0.4 

0.6 

0.7 

0.9 

.  i 

i.4 

1.6 

1.9 

2.2 

2.6 

2.9 

3.3 

3.7 

3  3o 

o.o 

o. 

o. 

0.2 

o.3 

o.5 

0.7 

0.9 

i.i 

.3 

1.6 

1.9 

2.2 

2.6 

3.o 

3.4 

3.8 

4.3 

4    o 

o.o 

o. 

0. 

0.2 

0.4 

o.5 

0.7 

I.O 

I  .2 

.5 

1.8 

2.2 

2.6 

3.0 

3.4 

3.9 

4.4 

4-9 

4   3o 

o.o 

0. 

O.2 

0.3 

0.4 

0.6 

0.8 

i  .1 

i.4 

•7 

2.  I 

2.5 

2.9 

3.3 

3.8 

4.4 

4.9 

5.5 

5     o 

0.0 

0. 

0.2 

0.3 

o.5 

0.7 

0.9 

I  .2 

i.5 

•9 

2.3 

2.7 

3.2 

3.7 

4.3 

4.8 

5.5 

6.1 

5  3o 

o.o 

0. 

O.2 

o.3 

o.5 

0.7 

.0 

1.3 

i-7 

.  i 

2.5 

3.o 

3.5 

4.i 

4-7 

5.3 

6.0 

6.7 

6     o 

o.o 

o. 

0.2 

0.4 

0.6 

0.8 

.  i 

i.4 

1.8 

2.3 

2.7 

3.3 

3.8 

4.4 

5.i 

5.8 

6.6 

7.3 

6  3o 

o.o 

0. 

O.2 

0.4 

0.6 

0.9 

.2 

1.6 

2.0 

2.5 

3.o 

3.5 

4.i 

4.8 

5.5 

6.3 

7-1 

8.0 

7     o 

o.o 

0. 

O.2 

0.4 

0.7 

.0 

.3 

!-7 

2.  I 

2.6 

3.2 

3.8 

4.5 

5.2 

5.9 

6.8 

7-7 

8.6 

7  3o 

0.0 

o. 

0.3 

0.4 

0.7 

.0 

.4 

1.8 

2.3 

2.8 

3.4 

4.i 

4.8 

5.5 

6.3 

7.2 

8.2 

9.1 

8     o 

0.0 

o. 

o.3 

o.5 

0.8 

.  i 

.5 

1.9 

2.4 

3.o 

3.6 

4.3 

5.i 

5.9 

6.8 

7-7 

8.7 

9-7 

8  3o 

0.0 

0. 

o.3 

o.5 

0.8 

.  i 

.6 

2.O 

2.6 

3.2 

3.9 

4.6 

5.4 

6.3 

7.2 

8.2 

9.2 

io.3 

9     ° 

o.o 

0.  I 

o.3 

o.5 

0.8 

.2 

•7 

2.2 

2.8 

3.4 

4.i 

4.9 

5.7 

6.6 

7.6 

8.6 

9.8 

10.9 

9  3o 

0.0 

0.  I 

o.3 

0.6 

0.9 

.3 

.8 

2.3 

2.9 

3.5 

4.3 

5.i 

6.0 

6.9 

8.0 

9.1 

io.3 

ii.  5 

IO       0 

O.O  0.  I 

o.3 

0.6 

0.9 

.3 

.8 

2.413.0 

3.7 

4.5 

5.4 

6.3 

7.3 

8.4 

9.6 

10.8  12.  I 

10   3o 

O.O  0.2 

o.3 

0.6 

i  .0 

.4 

•9 

2.5 

3.2 

3.9 

4.7 

5.6 

6.6 

7-7 

8.8 

IO.O 

n.3 

12.6 

II        0 

o.o 

O.2 

o.4 

0.7 

I  .0 

.5 

2.O 

2.6 

3.3 

4.i 

5.o 

5.9 

6.9 

8.0 

9.2 

io.5 

ii.  8 

13.2 

ii    3o 

o.o 

0.2 

0.4 

0.7 

I  .  I 

.5 

2.  I 

2.7 

3.4 

4.2 

5.2 

6.1 

7.2 

8.3 

9.6 

10.9 

12.3 

i3.8 

12       0 

0.0 

0.2 

0.4 

0.7 

I  .  I 

.6 

2.2 

2.8 

3.6 

4.4 

5.4 

6.4 

7.5 

8.7 

IO.O 

ii.  4 

12.8 

i4.4 

12     30 

0.  I 

0.2 

0.4 

0.7 

I  .  I 

.6 

2.2 

2.9 

3.7 

4.6 

5.6 

6.6 

7.8 

9.0 

io.4 

ii.  8 

i3.3 

14.9 

i3     o 

O.  I 

0.2 

0.4 

0.8 

1.2 

•7 

2.3 

3.i 

3.9 

4.8 

5.8 

6.9 

8.1 

9-4 

10.8 

12.2 

i3.8 

i5.5 

i3  3o 

O.  I 

O.2 

0.4 

0.8 

I  .2 

.8 

2.4 

3.2 

4.0 

5.o 

6.0 

7-i 

8.4 

9-7 

ii  .1 

12.6 

i4.3 

16.0 

i4     o 

O.  I 

0.2 

o.5 

0.8 

1.3 

.8 

2.5 

3.3 

4.2 

5.  i 

6.2 

7-4 

8.7 

IO.O 

n.  5 

i3.i 

i4.8 

16.6 

i4  3o 

0.  I 

0.2 

o.5 

0.8 

i.3 

•9 

2.6 

3.4 

4.3 

5.3 

6.4 

7.6 

8.9 

io.3 

11.9 

i3.5 

i5.3 

17.1 

16     o 

0.  I 

0.2 

o.5 

0.9 

i.4 

2.0 

2.7 

3.5 

4.5 

5.5 

6.6 

7-9 

9.2 

10.7 

12.3 

14.0 

i5.8 

17.7 

i5  3o 

0.  I 

O.2 

o.5 

0.9 

i.4 

2.O 

2.8 

3.6 

4.6 

5.6 

6.8 

8.1 

9.5 

II.  0 

12.7 

i4.4 

i6.3 

18.2 

16     o 

0. 

O.2 

o.5 

0.9 

i.5 

2.1 

2.9 

3.7 

4.7 

5.8 

7.0 

8.3 

9.8 

ii.  3 

i3.o 

14.8 

16.7 

18.7 

16  3o 

0. 

0.2 

o.5 

0.9 

i.5 

2.1 

2.9 

3.8 

4.8 

5.9 

7.2 

8.6 

IO.O 

ii.  6 

i3.4 

15.2 

17.2 

19.2 

17     o 

0. 

0.2 

o.5 

.0 

i.5 

2.2 

3.o 

3.9 

5.o 

6.  i 

7-4 

8.8 

io.3 

12.  O 

i3.8 

i5.6 

17.7 

19.8 

17  3o 

o. 

O.2 

0.6 

.0 

1.6 

2.3 

3.i 

4.0 

5.  i 

6.3 

7.6 

9.0 

10.6 

12.3 

i4.i 

16.0 

18.1 

20.8 

18     o 

o. 

o.3 

0.6 

.0 

1.6 

2.3 

3.i 

4.i 

5.2 

6.4 

7.8 

9.2 

10.9 

12.6 

i4.5 

16.4 

18.6 

20.8 

18  3o 

o. 

o.3 

0.6 

.0 

1.6 

2.4 

3.2 

4.2 

5.3 

6.6 

8.0 

9.5 

ii.  i 

12.9 

i4.8 

16.8 

19.0 

21.3 

19     o 

o. 

o.3 

0.6 

.1 

i-7 

2.4 

3.3 

4.3 

5.5 

6.7 

8.2 

9-7 

ii.  3 

13.2 

15.2 

17.2 

i9.5 

21.8 

19  3o 

0. 

o.3 

0.6 

.  I 

!-7 

2.5 

3.4 

4.4 

5.6 

6.9 

8.3 

9.9 

ii.  6 

i3.5 

i5.5 

17.6 

19.9 

22.2 

20       O 

0. 

o.3 

0.6 

.  I 

1.8 

2.5 

3.5 

4.5 

5.7 

7.0 

8.5 

10.  I 

11.9 

i3.7 

i5.8 

18.0 

2O.3;22.7 

20    30 

o. 

o.3 

0.6 

.  I 

1.8 

2.6 

3.5 

4.6 

5.8 

7-i 

8.7 

io.3 

12.  I 

i4.o 

16.1 

i8.3 

2O.7 

23.2 

21        O 

o. 

o.3 

0.7 

.2 

1.8 

2.6 

3.6 

4.7 

5.9 

7.3 

8.9 

io.5 

12.4 

a.  3 

i6.5 

18.7 

21  .  I 

23.6 

21     30 

o. 

o.3 

0.7 

.2 

1  9 

2.7 

3.6 

4.8 

6.0 

7-4 

9.0 

10.7 

12.6 

i4.6 

16.8 

19.1 

21.5 

24.1 

22       O 

o. 

o.3 

0.7 

.2 

1.9 

2.7 

3.7 

4.8 

6.2 

7.6 

9.2 

10.9 

12.8 

14.9 

17.1 

19.4 

21.9 

24.6 

22    3o 

o. 

o.3 

0.7 

.2 

1.9 

2.8 

3.8 

4.9 

6.3 

7-7 

9-4 

ii  .  i 

i3.o 

i5.i 

i7.4 

19.8 

22.3 

25.0 

23       0 

0. 

o.3 

0.7 

.3 

2.0 

2.8 

3.9 

5.o 

6.4 

7.8 

9.5 

ii.  3 

i3.3 

i5.4 

17.7 

20  »  i 

22.7 

25.4 

23  3o 

o. 

o.3 

0.7 

.3 

2.O 

2.9 

3.9 

5.i 

6.5 

8.0 

9-7 

ii.  5 

!3.6 

i5.7 

18.0 

20.  5 

23.1 

25.  9 

24     o 

o. 

o.3 

0.7 

.3 

2.0 

2.9 

4.o 

5.2 

6.6 

8.1 

9.9 

11.7 

i3.8 

i5.9 

i8.3 

20.8 

23.5 

26.3 

24   3o 

0. 

o.3 

0.7 

.3 

2  .  I 

3.o 

4.0 

5.3 

6.7 

8.2 

10.  0 

11.9 

13.9 

16.1 

i8.5 

21  .  I 

23.9 

26.7 

25       0 

0. 

o.3jo.7 

.3 

2  .  I 

3.o 

4.i 

5.3 

6.8 

8.4 

10.  I 

12.0 

i4.i 

16.4 

18.8 

21  .4 

24.2 

27.1 

25  3o 

o. 

o.3|o.8 

.3 

2.  I 

3.o 

4.2 

5.4 

6.9 

8.5 

io.3 

12.2 

i4.3 

16.6 

19.1 

21.7 

24.5 

27.5 

26     o 

0. 

o.3 

0.8 

.4 

2.2 

3.i 

4.3 

5.5 

7.0 

8.6 

10.4 

12.4 

i4.5 

16.8 

I9.3 

22  .O 

24.9 

27.9 

26  3o 

0. 

o.3 

0.8 

•  4 

2.2 

3.i 

4.3 

5.6 

7*1 

8.7 

10.6 

12.6 

i4.7 

17.1 

19.6 

22.3 

25.2 

28.2 

27     o 

0. 

o.3 

0.8 

.4 

2.2 

3.2 

4.4 

5.6 

7-i 

8.8 

10.7 

12.7 

14.9 

17.8 

19.9 

22.6 

25.5 

28.6 

27   3o 

o. 

o.4 

0.8 

•4 

2.2 

3.2 

4.4 

5.7 

7.2 

8.9 

10.8 

12.9 

1.5,.  i 

i7.5 

20.  i 

22.9 

25.8 

28.9 

28     o 

o. 

0.4 

0.8 

•4 

2.3 

3.3 

4.5 

5.8 

7.3 

9.0 

10.9 

i3.o 

i.5,3 

17.7 

20.4 

23.2 

26.2 

29.3 

28  3o 

o. 

0.4 

0.8 

i  P 

2.3 

3.3 

4.5 

5.9 

7-4 

9.1 

ii  .1 

13.2 

i5.5 

17.9 

20.6 

23.4 

26.5 

29.6 

29     o 

o. 

0.4 

0.8 

i  r 

2.3 

3.3 

4.5 

5.9 

7.5 

9.3 

II  .2 

i3.3 

i5.6  18.1 

20.8 

23.7 

26.8 

3o.o 

BB 


386 


TABLE   XIX. — SEMI-DIURNAL   ARCS. 


Latitude  of  the  Race. 

Star's 
Dec. 

40 

80  IQo 

120  |  140 

16o 

18o 

2Qo 

22o 

24o 

26o  J  28o  |  30o  |  320 

o 

h.  m. 

h.  m.  h.  m.\h.  m.\h.  m 

h.  m  h.  m. 

h.  m.  h.  m. 

h.  m. 

h.  m.\h.  m.  h.  m. 

h.  m. 

29  N. 

6  9 

6  18  6  22 

6  27 

6  32 

6  37 

6  4i 

6  47 

6  52 

6  57 

1  3 

7  87  16 

7  21 

28 

6  9 

6  17  6  21 

6  26 

6  3o 

6  35 

6  4o 

6  45 

6  5o 

6  55 

7  o 

7  6 

7  ii 

7  18 

27 

6  8 

6  166  21 

6  25 

6  29 

6  34 

6  38 

6  43 

6  48 

6  52 

6  58 

7  3 

7  8 

7  i4 

26 

6  8 

6  166  20 

6  24 

6  28 

6  32 

6  36 

6  4i 

6  45 

6  5o 

6  55 

7  o 

7  5 

7  ii 

25 

6  7 

6  i56  19 

6  23 

6  27 

6  3i 

6  35 

6  39 

6  43 

6  48 

6  53 

6  57 

7  2 

7  8 

24 

6  7 

6  146  18 

6  22 

6  25 

6  29 

6  33 

6  37 

6  41 

6  46 

6  5o 

6  55 

7  o 

7  5 

23 

6  7 

6  i4 

6  176  21 

6  24 

6  28 

6  32 

6  36 

6  39 

6  44 

6  48 

6  52 

6  57 

7  2 

22 

6  6 

6  i3 

6  1  6  6  20 

6  23 

6  27 

6  3o 

6  34 

6  38 

6  4i 

6  45 

6  5o 

6  54 

6  58 

21 

6  6 

6  12 

6  16  6  19  6  22 

6  25 

6  29 

6  32 

6  36 

6  39 

6  43 

6  47l6  5i 

6  56 

2O 

6  6 

6  iaj6  i5 

6  i8j6  21 

6  24 

6  27 

6  3o 

6  34 

b  37 

6  4i 

b  45jb  49 

b  53 

?9 

6  6 

6  ii  6  i4 

6  176  20 

6  23 

6  26 

6  29 

6  32 

6  35 

6  39 

6  42[6  46 

6  5o 

18 

6  5 

6  10  6  i36  166  19 

6  21 

6  24 

6  27 

6  3o 

6  33 

6  36 

6  4o 

6  43 

6  47 

J7 

6  5 

6  10  6  12 

6  i56  17 

6  20 

6  23 

6  26 

6  28 

6  3i 

6  34 

6  37 

6  4i 

6  44 

16 

6  5 

3  9612 

6  146  16 

6  19 

6  21 

6  24 

6  27 

6  29 

6  32 

6  35 

6  38 

6  4i 

i5 

6  4 

6  9 

6  ii 

6  i36  i5 

6  18 

6  20 

6  22 

6  26 

6  27 

6  3o 

6  33 

6  36 

6  39 

i4 

6  4 

6  8 

6  10 

6  12 

6  14 

6  16 

6  19 

6  21 

6  23 

6  25 

6  28 

6  3o 

6  33 

6  36 

i3 

6  4 

6  7 

6  9 

6  ii 

6  1  3 

6  i5 

6  17 

6  i9 

6  21 

6  24 

6  26 

6  28 

6  3i 

6  33 

12 

6  3 

6  7 

6  9 

6  16 

6  12 

6  i4 

6  16 

6  18 

6  20 

6  22 

6  24 

6  26 

6  28 

6  3i 

II 

6  3 

6  6 

6  8 

6  9J6  ii 

6  i3 

6  14 

6  16 

6  18 

6  20 

6  22 

6  24 

6  26 

6  28 

IO 

6  3 

6  6 

6  76  96  10 

6  12 

6  i3 

6  i5  6  iG 

b  18 

b  20 

6  22 

b  28 

b  25 

9 

6  3 

6  5 

6  6 

6  8 

6  9 

6  10 

6  12 

6  i3 

6  i5 

6  16 

6  18 

6  19 

6  21 

6  23 

8 

6   2 

6  5 

6  6 

6  7 

6  8 

6  9 

6  10 

6  12 

6  i3 

6  i4 

6  16 

6  17 

6  I9 

6  20 

7 

6   2 

6  4 

6  5 

6  6 

6  7 

6  8 

6  9 

6  10 

6  ii 

6  i3 

6  i4 

6  16 

6  16 

6  18 

6 

6   2 

6  3 

6  4 

6  5 

6  6 

6  7 

6  8 

6  9 

6  10 

6  ii 

6  12 

6  i3 

6  i4 

6  i5 

5 

6  i 

6  3 

6  4 

6  4 

6  5 

6  6 

6  7 

6  7 

6  8 

6  9 

6  10 

6  n 

6  12 

6  i3 

4 

6  i 

6   2 

6  3 

6  3 

6  4 

6  5 

6  5 

6  6 

6  6 

6  7 

6  8 

6  9 

6  9 

6  10 

3 

6  i 

6   2 

6   2 

6  3 

6  3 

6  3 

6  4 

6  4 

6  5 

6  5 

6  6 

6  6 

6  7 

6  8 

2 

6  i 

6  i 

6  i 

6   2 

6   2 

6   2 

6  3 

6  3 

6  3 

6  4 

6  4 

6  4 

6  5 

6  5 

i  N. 

6  o 

6  i 

6  i 

6  i 

6  i 

6  i 

6  i 

6  i 

6   2 

6   2 

6   2 

6   2 

6   2 

6   2 

o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

b  o 

6  o 

i  S. 

6  o 

5  59 

5  59 

5  59 

5  59 

5  59 

5  59 

5  59 

5  58 

5  58 

5  58 

5  58 

5  58 

5  58 

2 

5  59 

5  59 

5  59 

5  58 

5  58 

5  58 

5  57 

5  57 

5  57 

5  56 

5  56 

5  56 

5  55 

5  55 

3 

5  59 

5  58 

5  58 

5  57 

5  57 

5  57 

5  56 

5  56 

5  55 

5  55 

5  54 

5  54 

5  53 

5  62 

4 

59 

5  58 

5  57 

5  57 

5  56 

5  55 

5  55 

5  54 

5  54 

5  53 

5  52 

5  5i 

5  5 

5  5o 

5 

59 

5  57 

5  56 

5  56 

5  55 

5  54 

5  53 

5  53 

5  52 

5  5i 

5  5o 

5  49 

5  48 

5  47 

6 

58 

5  57 

5  56 

5  55 

5  54 

5  53 

5  52 

5  5i 

5  5o 

5  49 

5  48 

547 

5  46 

5  45 

7 

58 

5  ,56 

5  55 

5  54 

5  53 

5  52 

5  5i 

5  5o 

5  49 

5  47 

5  46 

5  45 

5  44 

5  42 

8 

58 

5  55 

5  54 

5  53 

5  52 

5  5i 

5  5o 

5  48 

5  47 

5  46 

5  44 

5  43 

5  41 

5  4o 

9 

57 

5  55 

5  54 

5  52 

5  5i 

5  5o 

5  48 

5  476  45 

5  44 

5  42 

5  41 

5  39 

5  37 

IO 

57 

5  54 

5  53 

5  5i 

5  5o 

5  48 

5  47 

5  45 

5  44 

5  42 

5  4o 

b  38 

5  3- 

5  35 

ii 

57 

5  54 

5  52 

5  5i 

5  49 

5  47 

5  46  5  44 

5  42 

5  4o 

5  38 

5  36 

5  34 

5  32 

12 

57 

5  53 

5  5i 

5  5o 

5  48 

5  46 

5  44 

5  42 

5  4o 

5  38 

5  36 

5  34 

5  32 

5  29 

i3 

56 

5  53 

5  5i 

5  49 

5  47 

5  45 

5  43 

5  41 

5  39 

5  36 

5  34 

5  32 

5  29 

5  27 

i4 

56 

5  52 

5  5o 

5  48 

5  46 

5  44 

5  4i 

5  39 

5  37 

5  35 

5  32 

5  3o 

5  27 

5  24 

i5 

56 

5  5i 

5  49 

5  47 

5  45 

5  42 

5  4o 

5  38 

5  35 

5  33 

5  3o 

5  27 

5  24 

5  21 

16 

5  55 

5  5i 

5  48 

5  46 

5  44 

5  41 

5  39 

5  36 

5  33 

5  3i 

5  28 

5  26 

5  22 

5  19 

ft 

5  55 

5  5o 

5  48 

5  45 

5  43 

5  4o 

5  37 

5  34 

5  32 

5  28 

5  26 

5  23 

5  19 

5  16 

18 

5  55 

5  5o 

5  47 

5  44 

5  4i 

5  39 

5  36 

5  33 

5  3o 

5  27 

5  24 

5  20 

5  17 

5  i5 

19 

5  54 

5  49 

5  46 

5  43 

5  4o 

5  37 

5  34 

5  3i 

5  28 

5  25 

5  21 

5  18 

5  i^ 

5  10 

20 

5  54 

5  48 

5  45 

5  42 

5  39 

5  36 

5  33 

5  3o 

5  26 

5  28 

5  19 

5  16 

5  ii 

5  7 

21 

5  54 

5  48 

5  44 

5  4i 

5  38 

5  35 

5  3i 

5  28 

5  24 

5  21 

5  17 

5  i3 

5  9 

5  4 

22 

5  54 

5  47 

5  44 

5  4o 

5  37 

5  33 

5  3o 

5  26 

5  22 

5  19 

5  i5 

5  10 

5  6 

5   2 

23 

5  53 

5  46 

5  43 

5  39 

5  36 

5  32 

5  28 

5  24 

5  21 

5  16 

5  12 

5  8 

5  3 

4  58 

24 

5  53 

5  46 

5  42 

5  38 

5  35 

5  3i 

5  27 

5  23 

5  19 

5  14 

5  10 

5  5 

5  o 

4  55 

25 

5  53 

5  45 

5  4i 

5  37 

5  33 

5  29 

5  25 

5  21 

5  17 

5  12 

5  7 

5  3 

4  58 

4  52 

26 

5  52 

5  44 

5  4o 

5  36 

5  32 

5  28 

5  24 

5  i9 

5  i5 

5  10 

5  5 

5  o 

4  55 

4  49 

27 

5  52 

5  44 

5  39 

5  35 

5  3i 

5  26 

5  22 

5  17 

5  12 

5  8 

5   2 

i  57 

$  52 

4  46 

28 

5  5i 

5  43 

5  39 

5  34 

5  3o 

5  25 

5  20 

5  i5 

5  10 

y  5 

5  o 

i  54 

i  49 

4  42 

29  S. 

5  5i 

5  42 

5  38 

5  33 

5  28 

5  23 

5  19 

5  i3 

5  8 

3 

i  57 

i  52 

i  45 

*  39j 

TABLE    XI X. — S E MI-DIURNAL   ARCS. 


387 


Latitude  of  the  Place. 

Star's 
Dec. 

34° 

36o 

38o 

4Qo 

42o 

440 

46o 

480 

50o 

520 

54o  56o 

580 

60o 

o 

A.  TO 

h.  m 

h.  m 

h.  m 

h.  m. 

h.  m. 

h.  m 

h.  m. 

h.  m 

A.  m 

A.  m.h.  m 

A.  m 

A.  m. 

29  N. 

7  28 

7  35 

7  43 

7  5i 

8   0 

8  10 

8  20 

8  32 

8  45 

9   I 

9199  4i 

IO  10 

io  55 

28 

7  24 

7  3i 

7  38 

7  46 

7  55 

8  4 

8  i4 

8  25 

8  37 

8  52 

9  8 

9  28 

9  53 

io  28 

27 

7  20 

7  27 

7  34 

7  4i 

7  49 

7  58 

8  7 

8  18 

8  3o 

8  43 

8  58 

9  z6 

9  39 

io  8 

26 

7  17 

7  23 

7  3o 

7  37 

7  44 

7  52 

8  i 

8  ii 

8  22 

8  35 

8  49 

9  5 

o  25 

9  5i 

25 

7  i3 

7  19 

7  25 

7  32 

7  39 

7  47 

7  55 

8  5 

8  i5 

8  27 

8  4o 

8  54 

9  i3 

9  35 

24 

7  10 

7  16 

7  21 

7  28 

7  35 

7  42 

7  5o 

7  59 

8  8 

8  19 

8  3i 

8  45 

9  2 

9  22 

23 

7  7 

7  12 

7  17 

7  23 

7  3o 

7  37 

7  44 

7  53 

8   2 

8  12 

8  23 

8  36 

8  5i 

9  9 

22 

7  3 

7  § 

7  i4 

7  J9 

7  25 

7  32 

7  39 

7  47 

7  55 

8  5 

8  i5 

8  27 

8  4i 

8  58 

21 

7  o 

7  5 

7  10 

7  16 

7  21 

7  27 

7  34 

7  4i 

7  49 

7  58 

8  8 

8  19 

8  32 

8  47 

2O 

6  57 

7  J 

7  6 

7  IJ 

7  i? 

7  22 

7  29 

7  35 

7  43 

7  5i 

8   o 

8  II 

8  .22 

8  36 

19 

6  54 

6  58 

7  2 

7  7 

7  12 

7  18 

7  24 

7  3o 

7  37 

7  45 

7  53 

8  3 

8  i4 

8  26 

18 

6  5i 

6  55 

6  5g 

7  3 

7  8 

7  i3 

7  19 

7  25 

7  3i 

7  38 

7  46 

7  55 

8  5 

8  17 

17 

6  48 

6  5i 

6  55 

6  59 

7  4 

7  9 

7  i4 

7  19 

7  25 

7  32 

7  4o 

7  48 

7.67 

8  8 

16 

6  45 

6  48 

6  52 

6  56 

7  o 

7  4 

7  9 

7  i4 

7  20 

7  26 

7  33 

7  4i 

7  4$ 

7  59 

i5 

6  42 

6  45 

6  48 

6  5a 

6  56 

7  ° 

7  4 

7  9 

7  i4 

7  20 

7  27 

7  34 

7  42 

7  5i 

i4 

6  39 

6  42 

6  45 

6  48 

6  52 

6  56 

7  o 

7  4 

7  9 

7  i4 

7  20 

7  27 

7  ,34 

7  42 

i3 

6  36 

6  3g 

6  42 

6  45 

6  48 

6  62 

6  55 

6  Sg 

7  4 

7  9 

7  i4 

7  20 

7  27 

7  34 

12 

6  33 

6  36 

6  38 

6  4i 

6  44 

6  47 

6  5i 

6.55 

6  59 

7  3 

7  8 

7  i3 

7  .20 

7  26 

II 

6  3o 

6  32 

6  35 

6  38 

6  4o 

6  43 

6  46 

6  5o 

6  54 

6  58 

7  2 

7  7 

7  .12 

7  19 

10 

6  27 

6  29 

6  32 

0  34 

b  37 

b  39 

6  42 

6  45 

6"  49 

6  52 

6  56 

7  i 

7  6 

7  ii 

9 

6  25 

6  26 

6  28 

6  3i 

6  33 

6  35 

6  38 

6  4i 

6  44 

6  47 

6  5o 

6  54 

6  59 

7  4 

8 

6  22 

6  23 

6  25 

6  27 

6  29 

6  3i 

6  33 

6  36 

6  39 

6  4i 

6  45 

6  48 

6  52 

6  56 

7 

6  19 

6  20 

6  22 

6  24 

6  25 

6  27 

6  29 

6  3i 

6  34 

6  36 

6  39 

6  42 

6  45 

6  49 

6 

6  16 

6  18 

6  19 

5  20 

6  22 

6  23 

6  25 

6  27 

6  29 

6  3i 

6  33 

6  36 

6  39 

6  42 

5 

6  i4 

6  i5 

6  16 

6  17 

6  18 

6  19 

6  21 

6  22 

6  24 

6  26 

6  28 

6  3o 

6  32 

6  35 

4 

6  ii 

612 

6  i3 

6  i3 

6  i4 

6  i5 

6  17 

6  18 

6  19 

6  21 

6  22 

6  24 

6  26 

6  28 

3 

6  8 

6  9 

5  9 

6  io 

6  ii 

6  12 

612 

6  i3 

6  14 

6  i5 

6  17 

6  18 

6  19 

6  21 

2 

/?  £? 

/? 

A    n 

fi  8 

fi   n 

^  _ 

\   T 

/? 

fi  ••('• 

fi   T/ 

iN. 

6  3 

6  3 

6  3 

6  3 

6  4 

6  4 

\J    O 

6  4 

9 

6  4 

)  1  0 

6  5 

/  I  O 

6  5 

)   II 

6  6 

J  {x 

6  6 

U   1». 

6  6 

U  I  i 

6  7 

o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

6  o 

o 

6  o 

6  o 

6  o 

6  o 

6  o 

i  S. 

5  57 

5  57 

5  57 

5  57 

5  56 

5  56 

5  56 

5  56 

55 

5  55 

5  54 

5  54 

5-54 

5  53 

2 

5  55 

5  54 

5  54 

5  53 

5  53 

5  52 

5  52 

5  5i 

5o 

5  5o 

5  49 

5  48 

5  47 

5  46 

3 

5  52 

5  5i 

5  5i 

5  5o 

5  49 

5  48 

5  48 

5  47 

46 

5  45 

5  43 

5  42 

5  4i 

5  39 

4 

5  49 

5  48 

5  47 

5  47 

5  46 

5  45 

5  43 

5  42 

4i 

5  39 

5  38 

5  36 

5  34 

5  32 

5 

5  46 

5  45 

5  44 

5  43 

5  42 

5  4i 

5  39 

5  38 

36 

5  34 

5  32 

5  3o 

5  28 

5  25 

6 

5  44 

5  42 

5  41 

5  4o 

5  38 

5  37 

5  35 

5  33 

3i 

5  29 

5  27 

5  24 

5  21 

5  18 

7 

5  41 

5  4o 

5  38 

5  36 

5  35 

5  33 

5  3i 

5  29 

26 

5  24 

5  21 

5  18 

5  i5 

5  ii 

8 

5  38 

5  37 

5  35 

5  33 

5  3i 

5  29 

5  27 

5  24 

21 

5  19 

5  io 

5  12 

5  8 

5  4 

9 

5  35 

5  34 

5  32 

5  29 

5  27 

5  25 

5  22 

5  19 

16 

5  i3 

5  io 

5  6 

5  i 

4  56 

10 

5  33 

5  3i 

5  28 

5  26 

5  23 

5  21 

5  18 

5  i5 

ii 

5  8 

5  4 

4  59 

4  54 

4  4cy 

ii 

5  3o 

5  28 

5  25 

5  22 

5  20 

5  17 

5  14 

5  10 

6 

5   2 

4  58 

4  53 

4  48 

4  4i 

12 

5  27 

5  24 

5  22 

5  19 

5  16 

5  i3 

5  9 

5  5 

i 

4  57 

4  52 

4  47 

4  4o 

4  34 

i3 

5  24 

5  21 

5  18 

5  i5 

5  12 

5  8 

5  5 

5  i 

4  56 

4  5i 

4  46 

4  4o 

4  33 

4  26 

i4 

5  21 

5  18 

5  i5 

5  12 

5  8 

5  4 

5  o 

4  56 

4  5i 

4  46 

4  4o 

4  33 

4  26 

4  18 

!5 

5  18 

5  i5 

5  12 

5  8 

5  4 

5  o 

4  56 

4  5i 

4  46 

4  4o 

4  33 

4  26 

4  18 

4  9 

16 

b  ib 

5  12 

b  8 

b  4 

5  o 

4  56 

4  5  1. 

4  46 

i  4o 

4  34 

4  27 

4  19 

4  ii 

4  i 

J7 

5  12 

5  g 

b  b 

5  i 

4  56 

4  5i 

4  46 

4  4i 

i  35 

4  28 

i  20 

4  12 

4  3 

3  5s 

18 

5  9 

5  5 

5  i 

4  57 

4  52 

4  47 

4  4i 

4  35 

4  29 

i.  22 

i  i4 

4  5 

3  55 

3  43 

r9 

5  6 

5   2 

4  58 

4  53 

4  48 

4  42 

4  36 

4  3o 

4  23 

4  i5 

4  7 

3  57 

3  46 

3  34 

20 

5  3 

4  59 

4  54 

4  4g 

4  43 

4  38 

4  3! 

4  25 

4  17 

4  9 

i.  o 

3  49 

3  3b 

3  24 

21 

5  o 

4  55 

4  5o 

4  45 

4  39 

4  33 

4  26 

4  19 

4  ii 

4  2 

3  52 

3  4i 

3  28 

3  i3 

22 

4  57 

4  52 

4  46 

4  4i 

4  35 

4  28 

4  21 

4  i3 

4  5 

3  55 

3  45 

3  33 

3  19 

3   2 

23 

4  53 

4  48 

4  43 

4  37 

4  3o 

4  23 

4  16 

4  7 

3  58 

3  48 

3  37 

3  24 

3  9 

2  5l 

24 

4  5o 

4  44 

4  39 

4  32 

4  25 

4  18 

4  10 

4  i 

3  52 

3  4i 

3  29 

3  i5 

2.58 

2  38 

25 

4  47 

4  4i 

4  35 

4  28 

4  21 

4  i3 

4  5 

3  55 

3  45 

3  33. 

3  20 

3  6 

2  47 

2  25 

26 

4  43 

4  37 

4  3o 

4  23 

4  16 

4  8 

3  59 

3  49 

3  38 

3  25 

1  II 

2  55 

2  35 

2  9 

27 

4  4o 

4  33 

4  26 

4  19 

4  ii 

4  2 

3  53 

3  42 

3  3o 

3  17 

'   2 

2  44 

2  21 

I  52 

28 

4  36 

i  29 

4  22 

4  i4 

4  5 

3  56 

3  46 

3  35 

3  23 

3  8 

2  52 

2  32 

2   7 

I  32 

29  S. 

4  32 

4  25 

4  17 

4  9 

4  o 

3  5o 

3  4o 

3  28 

3  i5 

2  59  2  4l  2  19 

i  5o 

i  5 

TABLE    XX. 


To  convert  Millimeters  into  English  Inches. 


Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

I 

0.0394 

59 

2.3229 

117 

4.6o64 

662 

26.0635 

720 

28.347o 

778 

3o.63o5 

2 

.0787 

60 

.3622 

118 

.6458 

663 

.1028 

72I 

.3864 

779 

.6698 

3 

.Il8l 

61 

.4oi6 

II9 

.685i 

664 

.  l422 

722 

.4257 

78o 

.7o92 

4 

.i575 

62 

.44io 

120 

.7245 

665 

.1816 

723 

.465i 

781 

.7486 

5 

.1969 

63 

.48o4 

i3o 

5.1182 

666 

.22O9 

724 

.5o45 

782 

•7879 

6 

.2362 

64 

•5l97 

i4o 

.5u9 

667 

.2608 

725 

.5438 

783 

.8273 

7 

.2756 

65 

.559i 

i5o 

.9o56 

668 

•2997 

•726 

.5832 

784 

^8667 

8 

.3i5o 

66 

.5985 

1  60 

6.2993 

669 

.339i 

727 

.6226 

785 

.9o6o 

9 

.3543 

67 

.6378 

170 

.693o 

67o 

.3784 

728 

.6620 

786 

.9454 

10 

.3937 

68 

.6772 

180 

•  7.o867 

671 

:  .4178 

729 

.7oi3 

787 

.9848 

ii 

.433i 

69 

.7166 

I9o 

.48o5 

672 

•  4572 

73o 

.74o7 

788 

3i  .0242 

12 

.4724 

70 

.7560 

200 

.8742 

673 

.4965 

73i 

,78oo 

789. 

.o635 

i3 

.6118 

7i 

.7953 

210 

8.2679 

674 

.5359 

732 

.8i94 

79° 

.  I029 

i4 

.55i2 

72 

.8347 

22O 

.6616 

675 

.5753 

733 

.8588 

79i 

.1423 

i5 

.5906 

73 

.874i 

230 

9.o553 

676 

•  6i47 

734 

.898s 

792 

.i8i7 

1.6 

.6299 

74 

.9i34 

24o 

•  449o 

677 

.654o 

735 

.9375 

793 

.2210 

J7 

.6693 

75 

.9528 

25o 

.8427 

678 

.6934 

736 

.9769 

794 

.2604 

18 

.7087 

76 

.9922 

260 

10.2364 

679 

.7328 

737 

29.oi63 

795 

;.2998 

J9 

.748o 

77 

3.o3i6 

270 

.63oi 

680 

.772I 

738 

.o556 

796 

,339i 

20 

.7874 

78 

.O7o9 

2§0 

H.0238 

681 

,8n5 

739 

.o95o 

797 

,3785 

21 

.8268 

79 

.iio3 

29O 

.4i75 

682 

.85o9 

74o 

.i344 

798 

•4i?9 

22 

.8662 

80 

.i497 

3oo 

.8112 

683 

.8902 

74x 

.i738 

799 

.4573 

23 

.9o55 

81 

.  i89o 

3io 

I2.2O49 

684 

.9296 

742 

.2l3l 

800 

.4966 

24 

.9449 

82 

.2284 

320 

•6987 

685 

.969o 

743 

.2525 

810 

3i.89o3 

25 

.9843 

83 

.2678 

33o 

.9924 

686 

27.0084 

744 

.29IO 

820 

32.284o 

26 

1.0236 

84 

.3071 

34o 

13.386! 

687 

•°477 

745 

.33i2 

83o 

'.6778 

27 

.o63o 

85 

.3465 

35o 

.7798 

688 

.o87i 

746 

.37o6 

84o 

33.o7i5 

28 

.1024 

86 

.3859 

36o 

i4.i735 

689 

.1265 

74? 

.4ioo 

85o 

.4652 

29 

.i4i8 

87 

.4253 

37o 

.5672 

69o 

.i658 

748 

.4494 

860 

;.8589 

3o 

.1811 

88 

.4646 

38o 

.96o9 

69i 

.2O52 

749 

.4887 

87o 

34;.  2526 

3i 

.22O5 

89 

,5o4o 

39o 

i5.3546 

692 

.2446 

75o 

.5281 

880 

.6463 

32 

.2599 

9° 

.5434 

4oo 

.7483 

693 

.2840 

75i 

.5675 

89o 

35<.o4oo 

33 

.2992 

91 

.5827 

4io 

16.  1420 

694 

.3233 

752 

.6068 

9oo 

.4337 

34 

.3386 

92 

.6221 

420 

.5357 

695 

.3627 

753 

.6462 

9io 

>8274 

35 

.3780 

93 

.66:5 

43o 

.9294 

696 

•  4O2I 

754 

.6856 

92O 

36  .  221  I 

36 

.4i73 

94 

.7oo9 

44o 

I7.3232 

697 

•  44i4 

755 

.7249 

93o 

,6i48 

3? 

.4567 

95 

.7402 

45o 

.7i69 

698 

.4808 

756 

.7643 

94o 

37!.oo85 

38 

.4961 

96 

.7796 

46o 

18.1106 

699 

.52O2 

757 

.8o37 

95o 

.4o23 

39 

.5355 

97 

.8i9o 

47o 

.5o43 

7oo 

.5596 

758 

.843i 

96o 

.7960 

4o 

.5748 

98 

.8583 

48o 

.898o 

7OI 

.5989 

759 

.8824 

97° 

38.i897 

4i 

.6142 

99 

.8977 

49o 

I9.29i7 

702 

.6383 

76o 

.92l8 

98o 

.5834 

42 

.6536 

IOO 

.937i 

5oo 

.6854 

7o3 

.6777 

761 

.96l2 

99° 

>977J 

43 

.6929 

101 

.9764 

5io 

2O.O79I 

7o4 

.7i7o 

762 

3o.ooo5 

IOOO 

39:.37o8 

44 

.7323 

102 

4.oi58 

520 

.47s8 

7o5 

.7564 

763 

.o399 

45 

•771? 

io3 

.o552 

53o 

.8665 

7o6 

.7958 

764 

.o793 

"Prnnnrf  innal 

46 

.8111 

io4 

.o946 

54o 

21  .2602 

7°7 

.835i 

765 

.n87 

Parts. 

4? 

.85o4 

io5 

.i339 

55o 

.6539 

•708 

.8745 

766 

.i58o 

Millim- 

English 

48 

.8898 

106 

.i733 

56o 

22.o477 

7°9 

.9i39 

767 

.i974 

eters. 

Inches. 

49- 

.9292 

107 

.2127 

57o 

•  44i4 

7io 

.9533 

768 

.2368 

O.I 

o.oo39 

5o 

.9685 

1  08 

.2520 

58o 

.835i 

7ii 

.9927 

769 

.276l 

O.2 

0.0079 

5x 

2.0079 

io9 

.2914 

59o 

23.2288 

7I2 

28.0320 

77° 

.3i55 

0.3 

0.0118 

52 

.0473 

no 

.33o8 

600 

.6225 

7i3 

.o7i4 

771 

.3549 

0.4 

0.0157 

53 

.0867 

III 

.3702 

610 

24.0162 

7i4 

.1108 

772 

.3942 

o.5 

0.0197 

54 

.1260 

112 

.4o95 

620 

•  4o99 

7i5 

.  i5oi 

773 

.4336 

0.6 

O.O236 

55 

,  i654 

n3 

.4489 

63o 

.8o36 

7i6 

.i895 

774 

.473o 

o.7 

0^.0276 

56 

.  2048 

n4 

.4883 

64o 

25.i973 

7i7 

,2289 

775 

.5i24 

0.8 

o.o3i5 

57 

.2441 

ii5 

.5276 

65o 

.59io 

7i8 

.2683 

776 

.55i7 

o.9 

o.o354 

58 

.2835 

116 

.567o 

660 

.9847 

7i9 

.3o76 

777 

.59ii 

I  .0 

o'.  o394 

One  millimeter  equals  o.o3937079  English  inch. 


TABLE    XXI. 


389 


To  con-vert  English  Inches  into  Millimeters. 


Englis 
Inches 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 

Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

English 
Inches. 

Millim- 
eters. 

0.  I 

2.54 

5.9 

149.86 

II.7 

297.17 

I7.5 

444.49 

23.3    |59i.8i 

29.1 

739.  i  3 

.2 

5.o8 

6.0 

l52.4o 

.8 

299.7 

.6 

447.  oL. 

.4 

594.35 

.2 

741.67 

.3 

7.62 

.1 

154.94 

•9 

302.25 

•7 

449.57 

.5 

596.89 

.3 

744.21 

.4 

10.  16 

.2 

i57.48 

12.  O 

304.79 

.8 

452.ii 

.6 

599.43 

.4 

746,76 

.5 

12.70 

.3 

160.02 

.1 

307.88 

•9 

454.65 

•7 

601  .97 

.5 

749.29 

.6 

i5.24 

.4 

162.  56 

.2 

309.3- 

18.0 

457.19 

.8 

6o4.5i 

.6 

76  i.  83 

•7 

17.78 

.5 

i65.io 

.3 

3l2.4l 

.1 

459,73 

•9 

607.05 

•7 

754.37 

.8 

20.32 

.6 

167.64 

.4 

3i4.95 

.2 

462.27 

24.0 

609.59 

.8 

766.91 

•9 

22.86 

•7 

170.  i  8 

.5 

317.49 

.3 

464.8i 

.  i 

612.  1  3 

•9 

769.46 

I.O 

25.  4o 

.8 

172.72 

.6 

320.  o3 

.4 

467..35 

.2 

614.67 

3o.o 

761.99 

.1 

27.94 

•9 

175.26 

•7 

322.  57 

.5 

469.89 

.3 

617.21 

.1 

764.53 

.2 

3o.48 

7.0 

177.80 

.8 

325.ii 

.6 

472.43 

.4 

619.  75 

.2 

767.07 

.3 

33.02 

.  i 

i8o.34 

•9 

327.  65 

«7 

474.97 

.5 

622.29 

.3 

769.61 

.4 

35.56 

.2 

182.88 

i3.o 

33o.i9 

.8 

477-5i 

.6 

624.83 

.4 

772.16 

.5 

38.io 

.3 

i85.42 

.1 

332.  73 

•9 

48o.o5 

•7 

627.37 

.5 

774.69 

.6 

4o.64 

.4 

187.96 

.2 

335.27 

19.0 

482.59 

.8 

629.91 

.6 

777.23 

•7 

43.i8 

.5 

190.60 

.3 

337.81 

.1 

485.i3 

•9 

632.45 

•7 

779.77 

.8 

45.72 

.6 

193.0^ 

•  4 

34o.35 

.2 

487.67 

26.0 

634.99 

.8 

782.31 

•9 

48.26 

•7 

i95.58 

.5 

342.89 

.3 

490.21 

.  i 

637.5S 

•9 

784.86 

2.0 

5o.8o 

.8 

198.12 

.6 

345.43 

.4 

492.76 

.2 

64o.o7 

3i.o 

787.39 

.  I 

53.34 

•9 

200.66 

•7 

347.97 

.5  1495.29 

.3 

642.61 

.  i 

789.93 

.2 

55.88 

8.0 

2O3.20 

.8 

35o.5i 

.6  1497-83 

.4 

645.i5 

.2 

792.47 

.3 

58.42 

.1 

206.7^ 

•9 

353.o5 

•7 

5oo.37 

.5 

647.69 

.3 

796.01 

.4 

60.96 

.2 

208.28 

14.0 

355.  59 

.8 

502.91 

.6 

65o.23 

.4 

797.66 

.5 

63.  5o 

.3 

210.82 

.  i 

358.13 

•9 

5o5.45 

•  7 

662.77 

.5 

800.09 

.6 

66.  o4 

.4 

2i3.36 

.2 

36o.67 

20.  o 

507.99 

.8 

655.3i 

.6 

802.63 

•7 

68.58 

.5 

215.90 

.3 

363.21 

.  i 

5io.53 

•9 

667.  85 

•  7 

806.  17 

.8 

71  .  12 

.6 

218.44 

.4 

365.  75 

.2 

5i3.o7 

26.0 

660.39 

.8 

807.71 

•9 

73.66 

•  7 

220.98 

.5 

368.29 

.3  j5i5.6i 

.  i 

662.93 

•9 

810.26 

3.o 

76.20 

.8 

223.52 

.6 

37o.83 

.4 

5i8.i5 

.2 

666.47 

32.0 

812.79 

.  i 

78.74 

•9 

226.06 

•7 

373.37 

.5 

520.69 

.3 

668.01 

.  i 

8i5.33 

.2 

81.28 

9.0 

228.60 

.8 

375.9i 

.6 

523.23 

.4 

670.55 

.2 

817.87 

.3 

83.82 

.  i 

23l.l4 

•9 

378.45 

•  7 

525.77 

.5 

673.09 

.3 

820.41 

.4 

86.36 

.2 

233.68 

i5.o 

380.99 

.8 

528.3i 

.6 

676.  63 

.4 

822.96 

.5 

88.90 

.3 

236.22 

.  i 

383.53 

•9 

53o.85 

.7 

678.17 

.5 

826.49 

.6 

91.44 

.4 

238.76 

.2 

386.o7 

21.0 

533.  39 

.8 

680.71 

.6 

828.03 

•7 

93.98 

.5 

24i.3o 

.3 

388.6i 

.1 

535.  93 

•9 

683.25 

•  7 

83o.56 

.8 

96.52 

.6 

243.84 

.4 

391.15 

.2 

538.47 

27.0 

685.79 

.8 

833.10 

•9 

99.06 

•7 

246.38 

.5 

393.69 

.3 

54i.oi 

.1 

688.33 

•9 

835.64 

4.o 

01.  60 

.8 

248.92 

.6 

396.23 

.4 

543.55 

.2 

690.87 

33.o 

838.i8 

.  i 

04.14 

•9 

25i.46 

•7 

398.77 

.5 

546.09 

.3 

693.4i 

.1 

840.72 

.2 

06.68 

10.  0 

254.oo 

.8 

4oi.3i 

.6 

548.63 

•4 

696.  95 

.2 

843.26 

.3 

09.22 

.1 

256.54 

fc*9 

4o3.85 

.7 

55i.i7 

f  r  r\ 

.5 

698.49 

.3 

846.80 

.4 

11.76 

.2 

259.08 

16.0 

406.39 

.8 

553.71 

.6 

701.  o3 

.5 

i4.3o 

.3 

261.62 

.1 

408.93 

•9 

556.25 

•7 

703.67 

.6 

i6.84 

.4 

264.16 

.2 

4n.47 

22.  O 

558.79 

.8 

706.1  i 

Parts. 

•  •7 

19.38 

.5 

266.  70 

.3 

4i4-oi 

.  I 

56i  .33 

•9 

708.66 

English 

Millim- 

.8 

21  .92 

.6 

269.24 

.4 

4i6.55 

.2 

563.87 

28.0 

711.19 

Inches. 

eters. 

•9 

24.46 

•7 

271.78 

.5 

419.09 

.3 

566.  4i 

.1 

713.73 

O.OI 

0.254 

5.o 

27.00 

.8 

274.32 

.6 

421  .63 

.4 

568.95 

.2 

716.27 

O.O2 

0.5o8 

.  i 

29.54 

•9 

276.  85 

•7 

424.17 

.5 

571.49 

.3 

718.81 

o.o3 

0.762 

.2 

32.  08 

1  1  .0 

279.39 

.8 

426.71 

.6 

574.03 

.4 

721.35 

o.o4 

1.016 

.3 

34.62 

.  i 

281.93 

•9 

429.25 

•  7 

576.57 

.5 

723.89 

0.06 

i  .270 

.4 

37.16 

.2 

284.47 

17.0 

43i.79 

.8 

679.11 

.6 

726.43 

0.06 

1.624 

.5 

39.70 

.3 

287.01 

.1 

434.33 

•9 

58i.65 

•  7 

728.97 

0.07 

1.778 

.6 

42.24 

.4 

289.55 

.2 

436.87 

23.0 

684.19 

.8 

73i.5i 

0.08 

2.032 

•7 

44.78 

.5 

292.09 

.3 

439.41 

.1 

586.  73 

•9 

734.  o5 

0.09 

2.286 

.8 

47.32 

.6 

294.63 

.4 

44i.95 

.2 

589.27 

29.0 

736.69 

O.  IO 

2.54o 

One  English  inch  equals  26.39964  millimeters 


390     TABLE  XXII. — ALTITUDES  WITH  THE   BAROMETER. 


PART  I. 

Argument,  the  observed  Height  of  the  Barometer  at  either  Station. 


Inches 

Feet.    Diff. 

Inches 

Feet. 

Diff. 

Inches. 

Feet. 

Diff. 

Inches.!   Feet. 

Diff. 

II  .0 

II.  I 
1  1.  2 

ii.  3 

ii.  4 

i396.9 
1633.3 
i867.6 
2099.9 
233o.i 

236.4 
234.3 

232.3 
230.2 

16.0 
16.1 
16.2 
i6.3 

16.4 

III86.3 
II349.I 

11671  .7 

ii83i.5 

162.8 
161.8 
160.8 
i59.8 

21  .O 
21  .  I 
21  .2 
21.3 

21.4 

18291  .O 

i84i5.i 
i8538.7 
18661.6 

18784.0 

124.  i 
123.6 
122.9 
122.4 

26.0 
26.  I 
26.2 
26.3 
26.4 

2397i.3 

24071  .2 
24170.7 
24269.8 

loo.S 

99.9 
99.5 
99.1 

0  0  Q  0 

T  ^R  K 

TOT    ft 

no  Q 

•  ii.  5 
ii.  6 
ii.  7 
ii.  8 
11.9 

2558.3 
2784.5 
3oo8  .  7 
323i.i 
345i.6 

22o  .  2 
226.2 
224.2 
222.4 
22O.5 

i6.5 
16.6 
16.7 
16.8 
i6.9 

11990.3 
12148.2 
I23o5.i 
12461  .0 
12616.  I 

i  \JO  •  O 

i57.9 
i56.9 
i55.9 
i55.i 

21.5 
21.6 

21.7 

21.8 
21  ,9 

18905.8 
19027.0 

19147-7 
19267.8 
19387.4 

1  £  i  •  O 

I  2  I  .  2 

120.7 
120.  I 
119.6 

26.5 
26.6 
26.7 
26.8 
26.9 

24368.6 
24467.0 
24565.1 
24662.7 
24760.0 

90  .  o 
98.4 
98.i 
97-6 
97.3 

12.0 
12.  I 
12.2 

367o.2 

3887.o 

4lO2.O 

218.6 
216.8 

2l5.O 

r»TQ  Q 

17.0 
17.1 
17.2 

12770.2 
12923  .5 
i3o75.8 

i54.i 
i53.3 
i52.3 

T^T   5 

22  .O 
22.  I 
22.2 

19506.4 
19624.9 
19742.9 

II9.0 
II8.5 
IlS.O 

27.0 
27.  I 
27.2 

24857.0 
24953.6 
25o49.8 

97.0 
96.6 
96.2 

12.3 
12.4 

43i5.3 
4526.9 

<£  1  O  «  G 
211.  6 

i7.3 

l3227.3 

i3377.9 

1  «J  1  •  vJ 

i5o.6 

22.3 
22  .4 

19860.3 
19977.2 

II7.4 
116.9 

27.3 
27.4 

25i45.7 

25241.2 

$:l 

12.5 
12.6 
12.  7 

12.8 

,12.9 

4736.7 
4g44  •  9 
5i5i.4 
5356.4 
5559.7 

209.8 
208.2 
2o6.5 

205.0 

2o3.3 

I7.5 
17.6 
17.7 
17.8 
17.  9 

i3527.6 
i3676.5 
13824.5 
i397i.7 
i4n8  .0 

149.7 

148.9 
i48.o 
147.2 
i46.3 

22.5 
22.6 
22.7 
22.8 
22.9 

20093.6 
20209.4 

20324.8 

20439.6 

2o554.o 

116.4 
n5.8 
ii5.4 
n4.8 

ii4.4 

27.5 
27.6 
27.7 
27.8 
27.9 

25336.4 

25431.2 

25525.7 
256i9.9 
257i3.7 

95.2 

94.8 
94.5 
94.2 

93.8 

i3.o 

l3'.2 

i3.3 
i3.4 

576i.4 
596i.6 
6i6o.3 
6357.5 
6553.2 

2OI  .7 
2OO.2 
198.7 
197.2 
I95.7 

18.0 
18.1 
18.2 
i8.3 

18.4 

14263.6 
i44o8.3 
14552.3 

14695.4 
i4837.8 

i45.6 
144-7 
i44.o 
i43.i 
142.4 

23.0 
23.1 
23.2 

23.3 

23.4 

20667.8 
20781  .  i 
20894.0 
21006.4 
2iii8.3 

n3.8 
ii3.3 
112.9 
1  12.4 
111.9 

28.0 
28.1 
28.2 
28.3 
28.4 

25807.1 
259oo.3 
25993.i 
26o85.  6 
26177.7 

93.4 
93.2 
92.8 
92.5 
92.  i 

i3.5 
i3.6 
i3.7 
i3.8 
i3.9 

6747.5 
694o.3 
7i3i.7 

7321.7 

75io.3 

i94.3 
192.8 
191.4 
190.0 

188.6 

i8.5 
18.6 

18.8 
i8.9 

14979.4 
i5i2o.3 
i526o.3 
i5399.7 
15538.3 

i4i.6 
140.9 
i4o.o 
139.4 
138.6 

23.5 
23.6 
23.7 
23.8 
23.9 

21229.7 
2i34o.6 
2i45i  .  i 
2i56i  .  i 
21670.6 

in.4 
110.9 
110.5 

IIO.O 

109.5 

28.5 
28.6 
28.7 
28.8 
28.9 

26269.6 
2636i.i 
26452.3 
26543.2 
26633.7 

91.9 
91  .5 
91.2 
90.9 
90.5 

14.0 
1  4.  i 
14.2 
i4.3 

7697.6 
7883.6 
8068.2 
825i.5 

187.3 

186.0 
184.6 
i83.3 

T  89  T 

I9.O 

19.1 

19.2 
I9.3 

i58i3!s 
i5949.8 
i6o85.5 

137-9 

137.1 

i36.5 
i35.7 

I  0  D  •  O 

24.0 
24.1 
24.2 
24.3 

21779.7 
21888.4 
21996.6 
22104.  3 

109.  1 

108.7 

108.2 

107.7 

TO1"?   *3 

29.0 
29.1 
29.2 
29.3 

26724.0 
26813.9 
26903.5 
26992.8 

90.3 
89.9 
89.6 
89.3 

i4.4 

8433.6 

1  OJt  m  i 

19.4 

16220  .5 

24.4 

222II  .6 

107.0 

29.4 

27081.9 

9.i 

i4.5 

86i4.4 

l8o.8 

ig.S 

i6354.8 

i34.3 

24.5 

223i8.4 

106.8 

29.5 

27170.6 

88.7 

QQ   X 

14.6 

879,4.0 

179.6 
T_6  o 

19.6 

16488.5 

T  *3r*   r\ 

24.6 

22424.8 

Tofi  0 

29.6 

27259.0 

CO  •  Z£ 

CQ  T 

14.7 

i4.8 
14.9 

8972.3 
9149.5 
9325.5 

170.3 
177.2 
176.0 

19.7 
19.8 
19.9 

16621  .4 
i6753.7 
i6885.3 

I  02  .  CJ 

1  32.  3 
i3i.6 

24.7 
24.8 

24.9 

2253o.8 
22636.4 
22741.5 

1  UU  »  (j 

io5.6 
io5.i 

29.7 
29.8 
29.9 

27347.1 
27434.9 
27522.5 

CO  •  1 

87.8 
87.6 

i5.o 
i5.i 

15.2 

i5.3 
i5.4 

9673!8 
9846.2 
10017.5 
10187.7 

174.8 
173.5 
172.4 
I7I.3 
I7O.2 

2O.  O 

2O.  I 
2O.  2 

20.  3 

20.4 

I7oi6.3 
I7i46.6 
i7276.3 
I74o5.3 
i7533.7 

iSi.o 
i3o.3 

129.7 

129.0 

128.4 

25.  O 
25.1 
25.2 

25.3 
25.4 

22846.3 
22950.  6 
23o54-4 
23157.9 
23261.0 

104.8 
io4-3 
io3.8 
io3.5 
io3.i 

3o.o 
3o.i 

30.2 

3o.3 
3o.4 

27609,7 
27696.6 

27783.3 
27869.7 
27955.7 

87.2 

86.  9 
86.7 
86.4 
86.0 

i5.5 
i5.6 
i5.7 
i5.8 
i5.9 
16.0 

io356.8 
io524.8 
10691  .8 
10857.7 

I  1022.5 
III86.3 

169.1 

168.0 
167.0 
i65.9 
i64.8 
i63.8 

20.5 
20.6 

20.7 

20.  8 
20.9 

21.0 

i766i.4 
i7788.6 
i79i5.  i 
i8o4i  .0 
18166.  3 
18291  .0 

127.7 
127.2 
126.5 
I25.9 

125.3 

124.7 

25.5 
25.6 
25.7 

25.8 

26.0 

23363.6 
23465.9 
23567.7 
23669.2 
23770.3 
23871.0 

IO2.6 
102.3 

101.8 
101  .5 

IOI.I 

100.7 

3o.5 
3o.6 
30.7 
3o.8 
3o.9 
3i.o 

28127.1 
28212.3 
28297.3 
28382.0 

28466.4 

85.8 
85.6 
85.2 
85.o 
84-7 
84.4 

TABLE   XXI I. — A LTITUDES   WITH  THE   BAROMETER.      391 


PART  II. 

Correction  due  to  T — T',  or  the  Difference  of  the  Temperatures  of  the  Barometers  at  the  two 

Stations. 


This  correction  is  Negative  ichen  the  temperature  at  the  upper  station  is  lowest,  and 


mce  versa. 


T—  T'. 

Correc- 
tion. 

T—  T'. 

Correc- 
tion. 

_     „,,      Correc- 
tion. 

„,    „,      Correc- 
tion. 

T—  T'. 

Correc-      ~     „, 
tion. 

Correc- 
tion. 

Fah't.       Feet. 

Fah't. 

Feet. 

Fah't.        Feet. 

Fah't.         Feet. 

Fah't. 

Feet.         Fah't. 

Feet. 

'     1° 

2.3 

l4° 

32.8 

27°       63.2 

4o°       98.6 

53° 

124.1       66° 

i54.5 

2 

4-7 

i5 

35.i 

28        65.5 

4i         96.0 

54 

126.4       67 

i56.8 

3 

7.0 

16 

37.5 

29      67.9 

42         98.3 

55 

128.7      68 

159.2 

4 

9.4 

17 

39.8 

3o        70.2 

43       100.7 

56 

iSi.i       69 

i6i.5 

5 

II.7 

18 

42.1 

3i         72.6 

44        108.0 

57 

i33.4      70 

i63.9 

6 

i9 

44.5 

33        74.9 

45        io5.3 

58 

i35.8      71 

166.2 

7 

l6;4 

20 

46.8 

33         77.3 

46        107.7 

59 

1  38.  i       72 

168.6 

8 

18.7 

21 

49.2 

34        79  -6 

47        no.o 

60 

i4o.4      73 

170.9 

9 

21  .  I 

22 

5i.5 

35        81.9 

48       112.  4 

61 

142.8      74 

i73.3 

IO 

23.4 

23 

53.8 

36        84.3 

49       114.7 

62 

i45.i      75 

175.6  . 

ii 

25.8 

24 

56.2 

37        86.6 

5o        117.0 

63 

i47-5      76 

177.9 

12 

28.1 

25 

58.5 

38        89.0 

5i         119.4 

64 

149.8      77 

i8o.3 

i3 

3o.4 

26 

60.9 

89        91.8 

52        121.7 

65 

l52.2          78 

182.6 

PART  III. 

PART 

IY 

Correction  due  to  the  Change 

of  Gravity  from  the  Latitude 

®  r-H 

PART 

V. 

of  45°  to  the  Latitude  of  the 

1-Jj 

Place 

of  Observation. 

?-,        -+^ 

g  S3 

| 

Correction  due  to  the  Height  of  the 

Positive  from  Lat.  0°  to  45°  ; 

1 

lower  Station. 

Negative  from  Lat.  45°  to  90°. 

sS 

* 

Always  Positive. 

Latitude. 

O  jj1 

1' 

Height  of  Barometer  at  lower  Station. 

GO 

1( 

)°   20° 

30o 

40o 

03    £ 

^ 

Jj 

j 

.s 

CO 

d 

.s 

App. 
Alt. 

8( 

10 

450 

H 

CO 

00 

r—  I 

o 

CQ 

«r-t 

CO 
CQ 

• 

CO 

App. 
Alt. 

70o 

60o 

50o 

Feet. 

Feet. 

Fe 

et. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet 

Feet. 

Feet 

F 

jet. 

Fee 

t. 

Feet. 

Feet. 

Fee 

jt. 

Feet. 

IOOO 

2.6 

2 

.5 

2  .O 

1.3 

o.5 

0 

2.5 

1.6 

1.3 

I  .0 

O. 

8 

0.6 

0.4 

0 

.2 

IOOO 

2OOO 

5.3 

5 

.0 

4.1 

2.6 

0.9 

O 

5.2 

3.1 

2.5 

2.O 

I  . 

5 

i  .  i 

0.7 

0 

.  3 

2OOO 

3ooo 

7-9 

7 

.5 

6.1 

4.0 

O 

7- 

I 

4.7 

3.8 

3.o 

2. 

3 

1.7 

I  .1 

O 

.5 

3ooo 

4ooo 

10.6 

10 

.0 

8.1 

5.3 

1$ 

O 

IO. 

\ 

6.3 

5.  i 

4.o 

3. 

i 

2.2 

I  .4 

O 

•7 

4ooo 

5ooo 

l3.2 

12 

.4 

IO.  I 

6.6 

2.3 

O 

18.7 

7.8 

6.4 

5.o 

3. 

8 

2.8 

1.8 

0 

.8 

5ooo 

6000 

i5.9 

i4 

•9 

12.2 

7-9 

2.8 

O 

16.7 

9-4 

7.6 

6.0 

4. 

6 

3.3 

2.  I 

.0 

6000 

7000 

i8.5 

.4 

14.2 

9.3 

3.2 

O 

19.9 

ii  .0 

8.9 

7- 

5.4 

3.9 

2.5 

.2 

7000 

8000 

21  .2 

r9 

•9 

16.2 

10.6 

3.7 

0 

23. 

i 

12.5 

10.2 

( 

Jr. 

6. 

2 

4-4 

2.8 

.3 

8000 

9000 

23.8 

22 

.4 

i8.3 

11.9 

O 

26.4 

i4.i 

ii.  4 

9- 

6. 

9 

5.o 

3.2 

.5 

9000 

IOOOO 

26.5 

24 

•9 

20.3 

18.2 

4^6 

o 

29.8 

i5.7 

12.7 

IO. 

7- 

7 

5.5 

3.5 

•  7 

IOOOO 

IIOOO 

29.1 

27 

.4 

22.3 

i4-6 

5  .  i 

o 

33. 

3 

17.2 

i4«  o 

ii  . 

8. 

5 

6.1 

8.9 

.8 

IIOOO 

I2OOO 

3i.8 

29 

•9 

24.4 

i5.9 

5.5 

o 

36. 

9 

18.8 

i5.3 

12. 

9- 

2 

6.6 

4.2 

2 

.0 

I2OOO 

iSooo 

34.4 

32 

.4 

26.4 

17.2 

6.0 

o 

4o. 

6 

20.4 

i6.5 

1C 

. 

IO. 

0 

7.2 

4.6 

2 

2 

18000 

i4ooo 

87.1 

34 

•9 

28.4 

i8.5 

6.4 

o 

44-4 

21.9 

17.8 

r4. 

IO. 

8 

7-7 

4-9 

2 

3 

i4ooo 

iSooo 

39.7 

37 

.3 

3o.4 

19.9 

6.9 

0 

48. 

3 

23.5 

19.1 

if 

. 

ii  . 

5 

8.3 

5.3 

2 

5 

i5ooo 

16000 

42.4 

39 

.8 

82.5 

21  .2 

7-4 

o 

52. 

3 

25.1 

20.3 

16. 

12. 

3 

8.8 

5.6 

2 

7 

16000 

17000 

45.o 

42 

.  3 

34.5 

22.5 

7.8 

o 

56.4 

26.6 

21.6 

15 

i3. 

i 

9-4 

6.0 

2 

8 

17000 

18000 

47.7 

44 

.8 

36.5 

23.8 

8.3 

o 

60. 

5 

28.2 

22.9 

18. 

18. 

8 

9.9 

6.3 

3 

0 

18000 

19000 

5o.3 

47 

c 

38.6 

25.2 

8.7 

o 

64. 

8 

29.8 

24.  1 

19.2 

i4. 

6 

io.5 

6.7 

3 

2 

19000 

2OOOO 

53.o 

4o 

'.S 

4o.e 

26.5 

9.2 

0 

69. 

2 

3i.3 

25.4 

20.2 

i5  . 

4 

II.  0 

7.0 

3 

3 

2OOOO 

2IOOO 

55.6 

52 

42.  e 

27.8 

9-7 

o 

78. 

G 

82.9 

26.7 

21  .2 

16. 

i 

ii.  6 

7-4 

3 

5 

21000 

22OOO 

58.3 

54.8 

44.  -3 

29.1 

10.  I 

o 

78. 

2 

34.5 

28.0 

22.2 

16. 

9 

12.  I 

7-7 

3 

7 

22OOO 

28000 

60.9 

57.3 

46.-; 

3o.5 

10.6 

o 

82. 

n 

86.0 

29.2 

23.2 

17. 

7 

12.  7 

8.1 

3 

8 

28000 

24000 

63.6 

5g.8 

48.-; 

81.8 

II  .0 

o 

87. 

G 

37.6 

80.5 

24.2 

18. 

5 

13.2 

8.4 

4 

0 

24000 

25ooo 

66.2 

J 

33.i 

ii.  5 

o 

92. 

5 

89.1 

81.825.2 

19. 

2 

i3.8    8.8 

4 

i 

25000 

392       TABLE    XXII I. — I NTERPOLATION    BY    DIFFERENCES. 


Parts 
ofthe 
Unit 
of 
Time. 

Bessel's  Coefficients  for 

Parts 
ofthe 
Unit 
of 

Time. 

Bessel's  Coefficients  for 

2d  Diff.     3d  Diff. 

4th  Diff.    5th  Diff. 

2d  Diff.    3d  Diff. 

4th  Diff. 

5th  Diff. 

t.t-l 
2  ' 

t.t—l.t—k 
~2   JP 

t...t—  2 
4  ' 

t,.*-* 

5 

t.t—  1 
2 

t.t-l.  t-4 
2    3  ' 

t  .  .  .  t—'2. 
4 

t  .  .  .  t—t 
5 

O.OI 

—  .OO495 

.00081 

.ooo83 

—  .OOO08 

0.5l 

—  .  12495 

—  .OOO42 

.02343 

.oooo5 

.02 

.00980 

.00157 

.ooi65 

.00016 

.52 

.12480 

.ooo83 

.02340 

.00009 

.o3 

.01455 

.00228 

.00246 

.0002  3 

.53 

.12455 

.00125 

.02334 

.00014 

.04 

.01920 

.00294 

.oo326 

.ooo3o 

.54 

.  I242O 

.00166 

.02327 

.00019 

.o5 

.02375 

.oo356 

.oo4o5 

,ooo36 

.55 

.i2375 

.00206 

,023i8 

.00023 

.06 

.02820 

.oo4i4 

.oo483 

.ooo43 

.56 

.12320 

.00246 

.O23o6 

.00028 

.07 

.o3255 

.00467 

.oo56o 

.ooo48 

.57 

.12255 

.00286 

.02293 

.OOO32 

.08 

.o368o 

.oo5i5 

.oo636 

.ooo53 

.58 

.  12180 

.oo325 

.02278 

.ooo36 

.09 

.04095 

.oo56o 

.00711 

.ooo58 

.59 

.  12095 

.oo363 

.02260 

.ooo4i 

.10 

.o45oo 

.00600 

.00784 

.ooo63 

.60 

.  I2OOO 

.oo4oo 

.02240 

.ooo45 

.  ii 

.o4895 

.oo636 

.oo856 

.00067 

.61 

.IlSgS 

.oo436 

.02218 

.00049 

.  12 

.o528o 

.00669 

.00927 

.00070 

.62 

.11780 

.00471 

.02194 

.ooo53 

.13 

.o5655 

.00697 

.00996 

.00074 

.63 

.n655 

.oo5o5 

.02169 

.ooo56 

.i4 

.06020 

.00723 

.01064 

.00077 

.64 

.  Il52O 

.oo538 

.O2l4l 

.00060 

.16 

.o6375 

.00744 

.on3o 

.00079 

.65 

.u375 

.00569 

.0211  I 

,ooo63 

.16 

.06720 

.00762 

.oiigS 

.00081 

.66 

.11220 

.00598 

.0208O 

.00067 

•i? 

.07055 

.00776 

.01259 

.ooo83 

.67 

.no55 

.00626 

.02046 

.00070 

.18 

.07380 

.00787 

.Ol32I 

.ooo85 

.68 

.10880 

.oo653 

.O20IO 

.00072 

.19 

.07695 

.00795 

.oi38i 

.00086 

.69 

.10695 

.00677 

.OI973 

.00075 

.20 

.08000 

.00800 

.01440 

.00086 

.70 

.  io5oo 

.00700 

.01934 

.00077 

.21 

.08295 

.00802 

.01497 

.00087 

•71 

.  10295 

.00721 

.OlSgS 

.00080 

.22 

.o858o 

.00801 

.oi553 

.00087 

.72 

.  10080 

.00739 

.oi85o 

.00081 

.23 

.08855 

.00797 

.01606 

.00087 

.73 

.o9855 

.00756 

,oi8o5 

.ooo83 

.24 

.09120 

.00790 

.oi658 

.00086 

•74 

.09620 

.00770 

.01758 

.00084 

.25 

.09375 

.00781 

.01709 

.ooo85 

.75 

.09375 

.00781 

.01709 

.ooo85 

.26 

.09620 

.00770 

.01758 

.00084 

.76 

.09120 

.00790 

.oi658 

.00086 

.27 

.09855 

.00756 

.oi8o5 

.ooo83 

•77 

.o8855 

.00797 

.01606 

.00087 

.28 

.  10080 

.00739 

.oi85o 

.00081 

.78 

.o858o 

.00801 

.oi553 

.00087 

.29 

.  10295 

.00721 

.01893 

.00080 

•79 

.08295 

.00802 

.01497 

.00087 

.3o 

.io5oo 

.00700 

.01934 

.00077 

.80 

.08000 

.00800 

.oi44o 

.00086 

.3i 

.  10695 

.00677 

.01973 

.00075 

.81 

.07695 

.00795 

.oi38i 

.00086 

.32 

.10880 

.oo653 

.02010 

.00072 

.82 

.07380 

.00787 

.Ol32I 

.ooo85 

.33 

.no55 

.00626 

.02046 

.00070 

.83 

.07055 

.00776 

.01259 

.ooo83 

.34 

,11220 

.ooSgS 

.02080 

.00067 

.84 

.06720 

.00762 

.01195 

.00081 

.35 

.ii375 

.00569 

.021  I  I 

.ooo63 

.85 

.o6375 

.00744 

.on3o 

.00079 

.36 

.Il52O 

.oo538 

.O2l4l 

.00060 

.86 

.06020 

.00723 

.01064 

.00077 

.37 

.n655 

,oo5o5 

.02169 

,ooo56 

.87 

.o5655 

.00697 

.00996 

.00074 

.38 

.11780 

.00471 

.02194 

.ooo53 

.88 

.o528o 

.00669 

.00927 

.00070 

.39 

.n895 

,oo436 

.02218 

.00049 

.89 

.04895 

.oo636 

.oo856 

.00067 

.4o 

.12000 

.oo4oo 

.02240 

.ooo45 

•9° 

.o45oo 

.00600 

.00784 

.ooo63 

.4i 

.12095 

.oo363 

.O226O 

.ooo4i 

.91 

.04095 

,oo56o 

.00711 

.ooo58 

.42 

.  12180 

.oo325 

.02278 

.ooo36 

.92 

.o368o 

.oo5i5 

.oo636 

.ooo53 

.43 

.12255 

.00286 

.02293 

.ooo32 

.9.3 

.o3255 

.00467 

.oo56o 

.ooo48 

•  44 

.  12320 

.00246 

,O23o6 

.00028 

.94 

.02820 

,oo4i4 

.oo483 

.ooo43 

.45 

.12375 

.00206 

.O23i8 

.00023 

.96 

.02375 

.oo356 

.oo4o5 

.ooo36 

.46 

.I242O 

.00166 

.02327 

.00019 

.96 

.01920 

.00294 

.oo326 

.ooo3o 

•47 

.12455 

.OOI25 

.02334 

.oooi4 

•97 

.oi455 

.00228 

.00246 

.OO023 

.48 

.12480 

.ooo83 

.02340 

.00009 

.98 

.00980 

.00157 

.ooi65 

.00016 

.49 

.12495 

.00042 

.02343 

.ooooS 

•99 

.00495 

.00081 

.ooo83 

.00008 

.5o 

—  .12500 

.00000 

.02344 

—  .00000 

i  .00 

—  .00000 

—  .  ooooo 

.ooooo 

.ooooo 

TABLE    XXII  I. — I  INTERPOLATION   BY   DIFFERENCES.      393 


Parts 
ofthe 
Unit 
of 
Time. 

Binomial  Coefficients  for 

Parts 
ofthc 
Unit 
of 
Time. 

Binomial  Coefficients  for 

2d  Diff.  |  3d  Diff. 

4th  Diff.    5th  Diff. 

2d  Diff.    3d  Diff. 

4th  Diff.    5th  Diff. 

t.t—  I 
2  ' 

t.t—l.t—z 

2  ~3~~ 

t  .  .  .  t—  3 
4 

t  .  .  t—  4 
5 

t.t—  1 
2 

t.t—  I  .t—2 
~2    §"' 

t.  ..  t—3 
4  ' 

t  .  .  .  t—  4 
5 

O.OI 

—  .OO495 

.oo328 

—  .00245 

.00196 

0.5l 

—  .12495 

.06206 

—  .03863 

.02696 

.02 

.00980 

.00647 

.00482 

.00384 

.52 

.  12480 

.O6l57 

.08817 

.02657 

.o3 

.01455 

.00955 

.00709 

.oo563 

.53 

.12455 

.06108 

.08769 

.026l5 

.04 

.01920 

.01254 

.00928 

.00735 

.54 

.  12420 

.o6o44 

.08717 

.02572 

.o5 

.O2375 

.oi544 

.OIl39 

.00899 

.55 

.12875 

.05981 

.03664 

.02528 

.06 

.02820 

.01824 

.oi34o 

.oio56 

.56 

.12820 

.o59l4 

.08607 

.02482 

.07 

.03255 

.02094 

.oi534 

.01206 

.57 

.  12255 

.o5842 

.o3549 

.02434 

.08 

.o368o 

.02355 

.01719 

.01848 

.58 

.12180 

.o5765 

.o3488 

.02886 

.09 

.04095 

.02607 

.01897 

.oi483 

.59 

.12095 

.o5685 

.08425 

.02886 

.10 

.o45oo 

.0285o 

.02066 

.01612 

.60 

.  I2OOO 

.o56oo 

.08860 

.02285 

.  ii 

.04895 

.o3o84 

.02228 

.01788 

.61 

.Il895 

.o55n 

.08293 

.02233 

.12 

.o528o 

.03309 

.02382 

.01849 

.62 

.11780 

.05419 

.08224 

.02180 

.13 

.o5655 

.o3525 

.02529 

.oigSS 

.63 

.n655 

.05322 

.o3i54 

.02125 

.i4 

.06020 

.03732 

.02669 

.02060 

.64 

.  Il520 

.05222 

.o3o8i 

.02071 

.i5 

.o6375 

.08981 

.02801 

.02157 

.65 

.11875 

,o5ii9 

.08007 

.020l5 

.16 

.06720 

.O4l22 

.02926 

.02247 

.66 

.II22O 

.o5oi2 

.02982 

.01958 

•J7 

.07055 

.o43o4 

.o3o45 

.02332 

.67 

.no55 

.04901 

.02855 

.01901 

.18 

.07380 

.04477 

,o3i56 

.02412 

.68 

.10880 

.04787 

.02777 

.oi844 

.19 

.07695 

.04643 

.08261 

.02485 

.69 

.  10695 

.04670 

.02697 

.01785 

.20 

.08000 

.04800 

.o336o 

.02554 

.70 

.  io5oo 

.o455o 

.02616 

.0!727 

.21 

.08295 

.04949 

.o3452 

.02617 

•?i 

.  10295 

.04427 

.02534 

.01668 

.22 

.o858o 

.©5091 

.o3538 

.02674 

.72 

.  10080 

.o43oi 

.O245i 

,0l6o8 

.23 

.08855 

.05224 

,o36i8 

.02728 

.78 

.o9855 

.04172 

.02868 

.oi548 

.24 

.09120 

.o535o 

.03692 

.02776 

•74 

.09620 

.o4o4o 

.02283 

.01488 

.25 

.09375 

.05469 

.03760 

.02820 

.75 

.09875 

.08906 

.02197 

.01428 

.26 

.09620 

.o558o 

.03822 

.02859 

.76 

.09120 

.08770 

.02111 

.01868 

.27 

.o9855 

.o5683 

.08879 

.02894 

•77 

.o8855 

.08681 

.O2O24 

.01808 

.28 

.  10080 

.05779 

.03980 

.02924 

.78 

.o858o 

.08489 

.01987 

.01247 

.29 

.  10295 

.o5868 

.08976 

.02950 

•79 

.08295 

.03346 

.01848 

.01187 

.3o 

.  io5oo 

.oSgSo 

.o4oi6 

.02972 

.80 

.08000 

.  08200 

.01760 

.01126 

.3i 

.  10695 

.06025 

.o4o52 

.02990 

.81 

.07695 

.o3o52 

.01671 

.01066 

.32 

.10880 

.06093 

.04082 

.o3oo4 

.82 

.07880 

.02903 

.01682 

.01006 

.33 

.no55 

.o6i54 

.o4io8 

.o3oi5 

.83 

.07055 

.02751 

.01498 

.00946 

.34 

.  II220 

.06208 

.04129 

.08022 

.84 

.06720 

.02598 

.oi4o3 

.00887 

.35 

.n375 

.06256 

.o4i45 

.08026 

.85 

.o6375 

.02444 

.01814 

.00828 

.36 

.  Il52O 

.06298 

.o4i56 

.08026 

.86 

.06020 

.02288 

.01224 

.00769 

.37 

.n655 

.o6333 

.o4i64 

.o3o23 

.87 

.o5655 

.02180 

.01184 

.00710 

.38 

.11780 

.o636i 

.04167 

.08017 

.88 

.o528o 

.01971 

.oio45 

.oo652 

.39 

.11895 

.06384 

.o4i65 

.08007 

.89 

.04895 

.01811 

.00955 

.00594 

.4o 

.  12000 

.o64oo 

.o4i6o 

.02995 

•9° 

.o45oo 

.oi65o 

.00866 

.oo537 

.4i 

.  12095 

.o64io 

.o4i5i 

.02980 

.91 

.04095 

.oi488 

.00777 

.oo48o 

.42 

.  I2l8o 

.o64i5 

.04188 

.02962 

.92 

.o368o 

.01825 

.00689 

.00424 

.43 

.  12255 

.o64i3 

,o4l2I 

.02942 

.93 

.o3255 

.01161 

.00601 

.00869 

.44 

.12320 

.06406 

.o4ioo 

.02919 

.94 

.02820 

.00996 

.oo5i3 

.00814 

.45 

.i2375 

.o6394 

.04076 

.02894 

.95 

.02875 

.00881 

.00426 

.00260 

.46 

.  12420 

.06876 

.04049 

.02866 

.96 

.01920 

.00666 

.00889 

.00206 

•47 

.12455 

.06352 

.o4oi8 

.02836 

•97 

.oi455 

,oo5oo 

.00264 

,ooi54 

.48 

.12480 

.o6323 

.08984 

.02804 

.98 

.00980 

.00888 

.00168 

.00102 

.49 

.12495 

.06289 

.08946 

.02770 

•99 

.oo495 

.00167 

.00084 

.ooo5o 

.5o 

—  .  i25oo 

.o625o 

—  .08906 

.02784 

I  .00 

—  .00000 

.00000 

—  .00000 

.00000 

394 


TABLE    XXIY 


Logarithms  of  the  Coefficients  for  Interpolation  by  BesseVs  Formula. 


Argument  for 
T=12  hours. 

Logarithms  or  the  Coefficients  lor 

First  Differences. 

Second  Differ- 
ences. 

Third  Differ- 
ences. 

Fourth  Differ- 
ences. 

Fifth  Differ- 
ences. 

h.    TO. 

0       0 

OC 

—  oc 

OC 

oc 

—  oc 

5 

7.84i6375 

7.53758ft 

6.75336 

6.76092 

5.75485ft 

10 

8.1426675 

7.  83556n 

7-o45i8 

7.o6o38 

6.o48i4ft 

i5 

8.3i87588 

8.0085971 

7.21195 

7.23484 

6.2i636ft 

20 

8.4436975 

8.i3o43ft 

7.32746 

7.358n 

6.33328ft 

25 

8.  54o6o75 

8.22423ft 

7.41482 

7.4533o 

6.422o4ft 

3o 

8.6197888 

8.3oo28w 

7-48434 

7.53o7i 

6.49292ft 

35 

8.6867355 

8.364o6ft 

7.54149 

7.59584 

6.55i42ft 

4o 

8.7447275 

8.4i887ft 

7.58957 

7.65i97 

6.6oo82ft 

45 

8.7958800 

8.4668271 

7.63o68 

7.70121 

6.64322ft 

5o 

8.84i6375 

8.5o935ft 

7.66626 

7.745oi 

6.68oo7ft 

55 

8.883o3o2 

8.54749/1 

7.69734 

7.78439 

6.7i239ft 

I       O 

8.9208188 

8.582ooft 

7.72467 

7.82013 

6.  74o95ft 

5 

8.95558o9 

8.6i346ft 

7.  74883 

7.85279 

6.7663ift 

10 

8.9877655 

8.64232ft 

7.77026 

7.88282 

6.7889ift 

1  5 

9.0177288 

8.66893ft 

7.78932 

7.91058 

6.8091271 

20 

9.0457575 

8.69358ft 

7.80628 

7-93636 

6.8272Ift 

25 

9.0720864 

8.7i65oft 

7.82138 

7.96039 

6.84342ft 

3o 

9.0969100 

8.7378971 

7.8348o 

7.98286 

6.85792n 

35 

9.1203911 

8.7579171 

7.84670 

8.00394 

6.87o89ft 

4o 

9.  1426675 

8.77670ft 

7.85722 

8.02377 

6.8824471 

45 

9.i638568 

8.79437ft 

7.86646 

8.04246 

6.8927oft 

5o 

9.1840602 

8.8no3ft 

7.8745i 

8.06011 

6.9oi75ft 

55 

9.2033653 

8.82676ft 

7-88i47 

8.07681 

6.9096871 

2       0 

9.2218487 

8.84i647i 

7.88739 

8.09264 

6.9i655ft 

5 

9.2395775 

8.85573ft 

7.89235 

8.io766 

6.92243ft 

10 

9.2566109 

8.  869ioft 

7.89637 

8.12194 

6.  92737ft 

i5 

9.2730013 

8.8817971 

7.89952 

8.13553 

6.93i4ift 

20 

9.2887955 

8.89386ft 

7.90183 

8.i4846 

6.93458ft 

25 

9.3o4o355 

8.  9o5347i 

7.90333 

8.i6o78 

6.936927^ 

3o 

9.3187588 

8.9i627ft 

7.90404 

8.i7253 

6.93845ft 

35 

9.3329992 

8.  92669ft 

7.90399 

8.i8375 

6.  93920ft 

4o 

9.3467875 

8.9366m 

7.9o3i9 

8.i9446 

6.  939i9ft 

45 

9.36oi5i4 

8.946o8ft 

7.90166 

8.20469 

6.  93842ft 

5o 

9.373u64 

8.955i2ft 

7.89942 

8.2i446 

6.9369271 

55 

9.3857o55 

8.96374« 

7.89646 

8.2238i 

6.93468ft 

3     o 

9.3979400 

8.  97i97ft 

7.89279 

8.232-74 

6.9317171 

5 

9.4098392 

8.97983ft 

7.88841 

8.24128 

6.9280171 

10 

9.4214211 

8.98733ft 

7.88333 

8.24944 

6.92359ft 

i5 

9.4327021 

8.9945071 

7.87753 

8.25724 

6.9i84sft 

20 

9.4436975 

9-ooi34ft 

7.87100 

8.26470 

6.  9i252ft 

25 

9.4544214 

9.  00787ft 

7.86374 

8.27i83 

6.9o586ft 

3o 

9.4648868 

9.oi4o9ft 

7.85573 

8.27864 

6.89843ft 

35 

9.4751060 

9.o2oo3ft 

7.84695 

8.285i4 

6.8902Oft 

4o 

9  .4850902 

9.o2570ft 

7.83737 

8.29134 

6.88n6ft 

45 

9.49485oo 

9.o3io9ft 

7.82697 

8.29725 

6.8712971 

5o 

9.5o43953 

9«o3623ft 

7.81572 

8.30289 

6.86o53ft 

55 

9-5i37354 

9.o4inw 

7.80357 

8.30826 

6.8488771 

4     o 

9.5228787 

9.o45  76ft 

7.79048 

8.3i336 

6.83624n 

TABLE   XXIV. 


Logarithms  of  the  Coefficients  for  Interpolation  by  BesseVs  Formula, 


Argument  for 
T=  12  hours. 

Logarithms  oCthe  Coefficients  for 

First  Differences. 

Second  Differ- 
ences. 

Third  Differ- 
ences. 

Fourth  Differ- 
ences. 

Fifth  Differ- 
ences. 

h.    TO. 

4.    o 

9.5228787 

9.0457671 

7.79048 

8.3l336 

6.83624?! 

5 

9.53i8336 

9.o5oi6?i 

7.7764I 

8.  81821 

6.8226171 

IO 

9.5406075 

9.0643471 

7.76128 

8.32282 

6.8079171 

i.5 

9.5492077 

9.o583o7i 

7.745o3 

8.327i8 

6.7920671 

20 

9.5576409 

9.06204/1 

7.72758 

8.33i3o 

6.7760071 

25 

9.5659i34 

9.0665671 

'7.7o883 

8.335i9 

6.  7666171 

3o 

9.574o3i3 

9.068887! 

7.68867 

8.33886 

6.7368on 

35 

9.5820002 

9.0720071 

•7.66696 

8.3423o 

6.7i5427i 

4o 

9.5898255 

9.0749272 

7.64355 

8.34553 

6.6923271 

45 

9.5975124 

9.0776471 

7.61825 

8.34854 

6.6673o7i 

5o 

9.6o5o655 

9.0801771 

7.59082 

8.35i34 

6.64oi4n 

55 

9.6124895 

9.0825271 

•7.56098 

8.35394 

6.6106671 

5     o 

9.6197887 

9.0846871 

7.52837 

8.35633 

6.578i8n 

5 

9.6269674 

9.086657* 

7.49256 

8.35853 

6.5426971 

IO 

9.6340292 

9.0884571 

7.4529-7 

8.36o52 

6.5o3i97i 

i5 

9.6409781 

9.0900771 

7.4o883 

8.36232 

6.4692371 

20 

9.6478175 

9.0915271 

7.35912 

8.36392 

6.4096871 

25 

9.65455o9 

9.0927971 

7.3o24o 

8.36533 

6.353i07i 

3o 

9.6611814 

9.0938871 

7.23655 

8.36655 

6.2873771 

35 

9.6677123 

9.0948171 

7.i583o 

8.36768 

6.  2092271 

4o 

9.6741464 

9.0955771 

7.06214 

8.36842 

6.n3i5n 

45 

9.6804866 

9.0961671 

6.  93779 

8.369.07 

6.9888671 

5o 

9.6867355 

9.0965771 

6.76212 

8.36964 

5.8i324n 

55 

9.6928959 

9.0968371 

6.46i34 

8.36982 

5.5i249n 

6     o 

9.6989700 

9.0969171 

QC 

8.36991 

—  oc 

5 

9.7049604 

9.0968371 

6.46i347i 

8.36982 

5.5i249 

10 

9.7108692 

9.0965771 

6.  7621271 

8.36954 

5.8i324 

i5 

9.7166988 

9.0961671 

6.9377971 

8.36907 

5.  08886 

20 

9.7224511 

9.09557/1 

7.0621471 

8  .  36842 

6.n3i5 

25 

9.7281282 

9.0948171 

7.  i583o7i 

8.36758 

6.20922 

3o 

9.7337321 

9.0938871 

7.2365571 

8.36655 

6.28737 

35 

9.7392646 

9.09279™ 

7.3024071 

8.36533 

6.353io 

4o 

9.7447275 

9.0915271 

7.3591271 

8.36392 

6.40968 

45 

9.7501225 

9.0900771 

7.4o8.83n 

8.36232 

6.46923 

5o 

9.75545i4 

9.0884571 

7.4529771 

8.36o52 

6.5o3i9 

55 

9.7607156 

9.086657! 

7.4925671 

8.35853 

6.54269 

7     o 

9.7659167 

9.0846871 

7.5283771 

8.35633 

6.578i8 

5 

9.7710664 

9.0825271 

7.5609871 

8.35394 

6.6io55 

IO 

9.7761360 

9.0801771 

7.5908271 

8.3.5i34 

6.64oi4 

i5 

9.7811568 

9.07764?! 

7.6182571 

8.34854 

6.6673o 

20 

9.7861202 

9.0749271 

7.6435571 

8.34553 

6.69232 

25 

9.7910275 

9.0720071 

7.6669671 

8.3423o 

6.71642 

3o    . 

9.7958800 

9.0688871 

7.6886771 

8.33886 

6.7368o 

35 

9.8006789 

9.0655671 

7.7088371 

8.335i9 

6.76661 

4o 

9.8o54253 

9.0620471 

7.7275871 

8.33i3o 

6.77600 

45 

9.8101205 

9.o583o7i 

7.745o37i 

8.32718 

6.792o6 

5o 

9.8i47654 

9.05434/1 

7.7612871 

8.32282 

6.8o79i 

55 

9.8193611 

9.0601671 

7.7764i7i 

8.3i82i 

6.82261 

8     o 

9.8239087 

9,0467671 

7.79o487i 

8.3i336 

6.83624 

396 


TABLE    XXIV. 


Logarithms  of  the  Coefficients  for  Interpolation  by  Bessel's  Formula. 


Argument  for 
T  =  12  hours. 

Logarithms  of  the  Coefficients  lor 

First  Differences. 

Second  Differ- 
ences. 

Third  Differ- 
ences. 

Fourth  Differ- 
ences. 

Fifth  Differ- 
ences. 

A.    m. 

8      o 

9.8239087 

9.o4576n 

7.7904871 

8.3i336 

6.83624 

5 

9.8284092 

9.041117* 

7.8o3577* 

8.3o826 

6.84887 

10 

9.8328636 

9.036237* 

7.815727* 

8.30289 

6.86o53 

i5 

9.8372727 

9.031097* 

7.826977* 

8.29725 

6.87129 

20 

9.84i6375 

9.0257071 

7.837377* 

8.29134 

6.  88116 

25 

9.8459589 

9-o2oo3n 

7.846957* 

8.285i4 

6.89020 

3o 

9.  8502377 

9.014097* 

7.855737* 

8.27864 

6.89843 

35 

9.8544747 

9.0078771 

7.86374n 

8.27i83 

6.90586 

4o 

9.8586709 

9.001347* 

7.871007* 

8.2647o 

6.91252 

45 

9.8628268 

8.9945071 

7-877537* 

8.25724 

6.91842 

5o 

9.8669434 

8.9873371 

7.883337* 

8.24944 

6.92359 

55 

9.8710213 

8.97983n 

7.888417* 

8.24128 

6  .92801 

9     ° 

9.8750613 

8.9719771 

7.892797* 

8.23274 

6.93171 

5 

9.8790640 

8.96374?* 

7.896467* 

8.2238i 

6.  93468 

10 

9.883o3o2 

8.9551271 

7.899427* 

8.2i446 

6.93692 

i5 

9.8869605 

8.9460871 

7.901667* 

8.20469 

6.93842 

20 

9.8908555 

8.9366171 

7.903197* 

8.19446 

6  .  93919 

25 

9.8947160 

8.9266971 

7.903997* 

8.i8375 

6.93920 

3o 

9.8985424 

8.9162771 

7.904047* 

8.i7253 

6.93845 

35 

9.9023354 

8.9053471 

7.90333?* 

8.i6o78 

6.93692 

4o 

9.9060955 

8.8938671 

7.901837* 

8.i4846 

6.93458 

45 

9.9098234 

8.8817971 

7.899527* 

8.13553 

6.93141 

5o 

9.9135195 

8  .8691071 

7.896377* 

8.  12194 

6.  92737 

55 

9.9171845 

8.8557371 

7.892357* 

8.io766 

6.92243 

IO       O 

9.9208187 

8.84i64n 

7.887397* 

8.09264 

6.9i655 

5 

9.9244229 

8.8267671 

7.881477* 

8.o768i 

6.90968 

IO 

9.9279973 

8.8no37* 

7.874517* 

8.06011 

6.90175 

i5 

9.9315426 

8.7943771 

7.866467* 

8.04246 

6.89270 

20 

9.9350592 

8.  7767071 

7.857227* 

8.o2377 

6.88244 

25 

9.9385475 

8.7579171 

7.846707* 

8.00394 

6.87089 

3o 

9.9420080 

8.7378971 

7.834807* 

7.98286 

6.85792 

35 

9.9454412 

8.7i65o7* 

7.821387* 

7.96039 

6.84342 

4o 

9.9488475 

8.69358?* 

7.806287* 

7.93636 

6.82721 

45 

9.9522272 

8.6689371 

7.789327* 

7.9io58 

6.80912 

5o 

9.9555809 

8.6423271 

7.770267* 

7.88282 

6.  -78891 

55 

9.9589088 

8.  6i3467* 

7.748837* 

7.85279 

6.7663i 

II       0 

9.9622115 

8.5820071 

7.724677* 

7.82013 

6.74095 

5 

9.9654892 

8.54749?* 

7.697347* 

7.78439 

6.7i239 

10 

9.9687423 

8.5093571 

7.666267* 

7.745oi 

6.6800-7 

i5 

9.9719713 

8.4668271 

7.63o68n 

7.7OI2I 

6.64322 

20 

9.9751764 

8.  4188771 

7.589577* 

7.65i97 

6.60082 

25 

9.9783581 

8.364o6n 

7.54149^ 

7.59584 

6.55i42 

3o 

9.9815166 

8.3002871 

7-48434n 

7.53o7i 

6  .49292 

35 

9-9846523 

8.2242371 

7.414827* 

7.4533o 

6.42204 

4o 

9.9877655 

8.i3o437* 

7.327467* 

7.358ii 

6.33328 

45 

9.9908566 

8.oo8597* 

7.211957* 

7.23484 

6.2i636 

5o 

9.9939258 

y.8355671 

7.o45i8n 

7.o6o38 

6.o48i4 

55 

9.9969736 

7.53758n 

6.753367* 

6.76092 

5.75485 

12       O 

o.ooooooo 

—  * 

QC 

-ce 

GC 

TABLE   XXY. 


397 


To  compare  the  Centesimal  Thermometer  with  Fahrenheit's. 


Centes 

Fahrenheit 

Cent.   Fahrenheit. 

Cent,  j  Fahrenheit. 

Cent.  Fahrenh't 

Cent.  Fahrenh't. 

o 

o 

O 

o 

o 

o 

0 

0 

0 

o 

+  IOO 

+  212.0 

+  71 

+  169.8 

+42 

+  107.6 

+  i3 

+55.4 

—  16 

+     3.2 

99 

210.2 

7° 

1  58.o 

4i 

105.8 

12 

53.6 

—  17 

+    i«4 

98 

208.4 

69 

166.2 

4o 

io4.o 

II 

61.8 

—  18 

—  0.4 

.  97 

206.6 

68 

i54.4 

39 

I  O2  .  2 

10 

5o.o 

—  19 

—  •     2.2 

96 

204.8 

67 

162.6 

38 

100.4 

9 

•    48.2 

—  20 

—  4.0 

96 

203.0 

66 

i  5o.8 

37 

98.6 

46.4 

—  21 

—  5.8 

94 

201  .2 

65 

149.0 

36 

96.8 

7 

•  44.6 

—  22 

-   7.6. 

93 

199.4 

64 

l47-2 

35 

96.0 

6 

42.8 

—  23 

-  9-4. 

Proportional 

Parts 

92 

197.6 

63 

i45.4 

34 

93.2 

5 

.  4i  .0 

o/ 

—  II  .2 

-24 

c. 

Fah't. 

91 

I95.8 

62 

i43.6 

33 

91.4 

4 

39.a 

—  25 

—  i3.o  . 

0 

9° 

194.0 

-  61 

i4i.8 

32 

•       89.6 

3 

37.4 

-26 

—  14.8. 

O.  I 

0.18 

89 

192.2 

60 

i4o.o 

3i 

87.8 

2 

35.6 

—  27 

—  16.6 

0.2 

0.36 

88 

190.4 

-  69 

i38.2 

-  3o 

.     86.0 

+    I 

33.8 

—  28 

—  18.4 

o.3 

0.54 

87 

188.6 

•  58 

i36.4 

29 

84.2 

0 

32.0 

—  29 

—  2O.  2 

0.4 

0.72 

86 

186.8 

57 

i34.6 

•  28 

82.4 

—    I 

,  3o.2 

-3o 

—  22  .O 

0.5 

o.oo 

85 

186.0 

56 

i32.8 

27 

80.6 

—    2 

28.4 

—  3i 

—23.8 

0.6 

y 
i.  08 

84 

i83.2 

55 

i3i.o 

26 

78.8 

—  3 

26.6 

—  32 

—  26.6 

0.7 

1.26 

83 

181.4 

54 

129.2 

25 

77-° 

-  4 

24.8 

—33 

—27.4 

0.8 

i.44 

82 

179.6 

53 

127.4 

.  24 

76.2 

-  5 

23.0 

—34 

—  29.2 

O.Q 

i  .62 

81 

177.8 

62 

126.6 

•    23 

•     73.4 

—  6 

21  .2 

—35 

-3i.o 

y 
i  .0 

i.  80 

80 

176.0 

5i 

123.8 

22 

71.6 

—  7 

i9.4 

—36 

—  32.8 

79 

174.2 

5o 

122.  O 

21 

69.8 

—  8 

17.6 

-37 

—  34.6 

78 

I72.4 

49 

I2O.2 

20 

68.0 

—  9 

i5.8 

—38 

—  36.4 

77 

i7o.6 

48 

Il8.4 

•    19 

66.2 

—  10 

i4.o 

-39 

—  38.2 

76 

168.8 

47 

II6.6 

18 

64.4 

—  ii 

12.2 

—  4o 

—  4o.o 

75 

i67.o 

46 

n4.8 

17 

62.6 

—  12 

.    10.4 

-4  1 

—  4i.8 

74 

166.2 

45 

n3.o 

16 

60.8 

—  13 

.    8.6 

-42 

—43.6 

73 

i63.4 

44 

112.  2 

i5 

59.o 

—  14 

.    6.8 

-43 

-45.4 

+  72 

+  161.6 

+43 

+  109.4 

+  i4 

+  57.2 

—  15 

+  5.o 

-44 

—47.2 

x°  Centesimal=(32°+-:r0)  Fahrenheit, 
o 

TABLE    XXVI. 

To  compare  Reaumur's  Thermometer  with  Fahrenheit's. 


Reaum. 

Fahrenheit. 

R'm'r. 

Fahrenheit. 

R'm'r. 

Fahrenheit. 

R'm'r. 

Fahrenh't. 

R'm'r. 

Fahrenh't. 

0 

0 

0 

0 

0 

0 

o 

O 

o 

0 

+    80 

+  212.0 

+57 

+  160.26 

+  34 

+108.5 

+11 

+  56.75 

12 

+  5.o 

79 

209.75 

56 

168.0 

33 

106.26 

IO 

54.5 

—  13 

+  2.76 

78 

207.6 

55 

i55.75 

32 

io4.o 

9 

62.26 

—  14 

+  0.5 

77 

2O5.25 

54 

i53.5 

3i 

•     101.76 

8 

5o.o 

—  16 

—  1.76 

/  / 

76 

2O3.0 

53 

r 

161.26 

/ 

3o 

99.5 

7 

47.75 

/  r      r 

—  16 

-  4.0 

6         c 

Proportional 
Parts. 

75 

200.76 

o     c 

62 

r 

149.0 

//»               £ 

29 

o 

97.25 

6 

45.5 

T   Q 

8C 

R. 

Fah't. 

74 

I98.5 

61 

i46.75 

2o 

96  .O 

43  .  20 

o 

o 

73 

i96.25 

5o 

i44.5 

•    27 

92.76 

4 

4i  «o 

—  19 

—  10.76 

0.  I 

0.226 

72 

194.0 

•  49 

142.26 

26 

-        90.5 

3 

.  38.  76 

•^-20 

—  i3.o 

O.2 

0.45 

71 

I9i.75 

•  48 

i4o.o 

25 

88.25 

2 

36.5 

—  21 

—16.26 

o.3 

0.676 

7° 

i89.5 

•  47 

i37.75 

24 

86.0 

+     I 

34.25 

—  22 

-17.5 

0.4 

0.90 

69 

187,26 

•   46 

i35.5 

23 

83.  76 

O 

32.0 

—23 

—19.76 

0.5 

1  .126 

68 

186.0 

.45 

i33.25 

22 

8i.5 

—    I 

29.76 

-24 

—  22  .O 

0.6 

1.35 

67 

182.76 

•  44 

i3i.o 

21 

79.26 

—    2 

27.6 

—26 

—  24.25 

0.7 

1.676 

66 

180.6 

43 

128.76 

20 

77.0 

—  3 

26.26 

—26 

—  26.5 

0.8 

i.  80 

65 

178.26 

•  42 

126.6 

'9 

74.75 

-  4 

23.0 

—27 

—  28.76 

0.9 

2.026 

64 

176.0 

•  4i 

124.26 

18 

72.6 

—  5 

20.76 

—28 

—  3i.o 

I.O 

2.26 

63 

i73.75 

4o 

122.  O 

17 

70.26 

—  6 

i8.5 

—29 

—  33.25 

62 

171  .5 

39 

119.76 

16 

68.0 

—  7 

16.26 

—  3o 

—35.5 

61 

169.26 

38 

II7.5 

i5 

66.76 

—  8 

i4.o 

—  3i 

-37.76 

60 

167.0 

37 

II5.25 

i4 

63.5 

—  9 

11.76 

—  32 

—  4o.o 

59 

164.76 

36 

n3.o 

i  a 

61.26 

—  10 

9.5 

—33 

—  42.26 

+  58 

+  162.6 

+  35 

+  110.76 

+  12 

+  69.0 

—  ii 

+  7-26 

-34 

-44.5 

x°  Reaumur =(32°+-:r0)  Fahrenheit. 


398 


TABLE    XXVII. 


Height  of  Barometer  corresponding  to  Temperature  of 


Fah't 
Degrees. 

English 
Inches. 

Diff. 

Fah't      English 
Degrees.    Inches. 

Diff. 

Fah't 
Degrees. 

English 
Inches. 

Diff. 

Fah't 
Degrees. 

English 
Inches. 

Diff. 

i85.o 
.  I 

.2 

.3 
.4 
.5 
.6 

'.8 
•9 

.086 

.123 

.161 
.198 
.236 

.273 

.3io 

.348 
.386 

.o37 
.o37 
.o38 
.o37 
.o38 
.o37 
.o37 
.o38 
.o38 

^QQ 

190.8 

•9 

191.0 
.1 
.2 

.3 
.4 
.5 
.6 
•7 

19.326 
.368 
.409 
.45o 
.492 
.534 
.575 
.6i7 
.659 

.7OI 

.042 

.o4i 
.042 

.  O42 

.042 

.  O42 

.042 

196.6 

•7 

.8 

•9 
197.0 
.  i 

.2 

.3 

.4 
.5 

21.  85  1 

.897 
.943 
.989 

22.035 

.081 
.128 

.174 

.221 

.267 

.o46 
.o46 
.o46 
.o46 

.047 
.o46 

.047 
.o46 

202.4 

.5 
.6 

'.8 
•9 

203.0 

.  i 

.2 

.3 

24.647 
.697 

•  748 

•799 
.85o 
.901 
.952 
25.oo3 
.o55 
.106 

.o5o 
.o5i 
.061 
.o5i 
.o5i 
.061 
.o5i 

.052 

.o5i 

186.0 
.  i 

.2 

.3 
.4 
.5 
.6 

'.8 

187.0 
.  i 

.2 

.3 
•  4 
.5 
.6 

.8 

•9 

188.0 
.  i 

.2 

.3 
.4 
.5 
.6 

'.8 

•9 
189.0 
.1 

.2 

.3 
.4 
.5 
.6 

'.8 

•9 
190.0 
.  i 

.2 

.3 
•  4 
.5 
.6 

•  424 
.462 
.5oo 
.538 
.576 
.6i5 
.653 
.691 

;768 

.8o7 
.846 
.884 
.923 
.962 
18.001 
.o4o 

•°79 
.118 

.i58 

•197 
.236 

.276 

.3i5 
.355 
.395 
.434 

•  474 
.5i4 
.554 
.594 
.634 
•  674 
.7i4 
.755 

•795 
.835 
.876 
.917 

•957 
.998 
19.039 
.080 

.  121 
.  162 
.203 

.244 
.285 
.326 

.o38 
.o38 
.o38 
.o38 
.o39 
.o38 
.o38 
.o39 
.o38 
.039 
.o39 
.o38 
.o39 
.039 
.039 
.039 
.039 
.039 
.o4o 
.039 
.039 
.o4o 

.039 

.o4o 
.o4o 
.039 

.o4o 

.o4o 
.o4o 
.o4o 
.o4o 
.041 
.o4o 
.o4o 
.o4i 
.041 
.o4o 
.o4i 

.o4i 
.o4i 

.  o4i 
.  o4i 

.8 

•9 
192.0 
.  i 

.2 

o 

.  0 

.4 
.5 
.6 

•7 

.8 

193.0 
.  i 

.2 

.3 
.4 
.5 
.6 

'.8 

•9 
194.0 
.  i 

.2 

.3 

.4 
.5 
.6 

'.8 

195.0 
.  i 

.2 

.3 
.4 
.5 
.6 

•  1 
.8 

•9 
196.0 

.2 

.3 

.4 
.5 
.6 

.743 
.785 
.827 
.869 
.912 
.954 
.996 
20.039 
.082 
.124 
.  167 
.210 
.253 
.296 
.339 
.382 
.426 

.'5l2 

.556 

•599 
.643 
.687 
.73i 
.775 
.819 
.863 

•9°7 
.95i 
.996 

21  .o4o 
.084 
.  129 

.i74 
.218 
.263 
.3o8 
.353 
.398 
.443 
.488 
.533 
.578 
.623 
.669 
.7i4 
.76o 
.806 
.85i 

.  O42 
.O42 
.O42 

.o43 
.042 

.043 

.o43 
.042 
.o43 
.o43 
.o43 
.o43 
.o43 
.o43 
.o44 
.o43 
.o43 
•  o44 
.o43 
.o44 

.'o44 
.o44 
•  o44 
.o44 

•  o44 
.o45 

.o44 
.o45 
.o45 
.o44 
.o45 
.o45 
.o45 
.o45 
.o45 
.o45 
.o45 
.o45 
.o45 
.o46 
.o45 
.o46 
.o46 
.o45 

.6 

.*8 

•9 
198.0 
.  i 

.2 

.3 
.4 
.5 
.6 

•  7 
.8 

•9 
199.0 
.  i 

.2 

.3 
.4 
.5 
.6 

'.8 

•9 
200.  o 
.1 

.2 

.3 
.4 
.5 
.6 

it 

•9 

201  .0 
.1 
.2 

.3 
.4 
.5 
.6 

'.8 
•9 

202.  O 
.  I 
.2 
.3 

.4 

.3:4 
.$61 
.407 
•  454 
.$01 

.548 
.595 
.642 
.689 
.736 

!83i 

.879 
.926 
•974 

23.  O22 

.o7o 

.118 
.166 
.214 

.262 

.3n 
.359 
•  4o7 
.456 
.5o5 
.553 
.602 
.65i 
.700 

.798 
•  847 
.897 
•  946 
.996 
24  .  o45 
.095 
.i45 
.i95 
.245 
.295 
.345 
.395 
.445 
.495 
.546 
.596 
.64? 

.o47 
.046 
.047 
.047 
•  047 
.o47 
.047 
.047 
.047 
.048 
.o47 
.048 
.047 
.o48 
.o48 
.o48 
.048 
.048 
.048 
.048 
.049 
.o48 
.o48 
.049 
.049 
«o49 
.049 
.049 
.049 
.049 
.049 
.049 
.o5o 
.049 
.o5o 
.049 
.o5o 
.o5o 
.o5o 
.o5o 
.o5o 
.o5o 
.o5o 
.o5o 
.o5o 
.o5i 
.o5o 
.o5i 

.4 
.5 
.6 

•7 
.8 

204.0 
.  I 
.2 

0 

•  4 
•7 

205.0 

.  I 

.2 

.3 
.4 
.5 
.6 

.*8 

206.0 
.1 

.2 

.3 
.4 
.5 
.6 

.8 

•9 
207.0 

.  i 

.2 

.3 
.4 
.5 
.6 

.'8 

•9 

208.0 
.  i 

.2 

.i58 

.210 
.26l 

.3i3 
.365 
.4i7 

!s2? 

.573 
.626 

.678 
.73o 
.783 
.836 
.888 

.994 
26.047 

.100 

.i53 
.206 
.259 
.3i3 
.366 
.420 
.473 
.527 
.58i 
.635 
.689 
.743 

•797 
.852 
.906 
.961 
27.015 
.070 

.125 

.180 

.235 
.290 

.345 

.*456 
.5n 
.566 
.622 
.678 
.733 

.o5i 

.052 
.052 
.052 
.052 

.o52 

.052 

.o53 

.052 

.o52 
.o53 

.o53 

.052 

.o53 
.o53 
.o53 
.o53 
.o53 
.o53 
.o53 
.o54 
.o53 
.o54 
.o53 
.o54 
.o54 
.o54 
.o54 
.o54 
.o54 
.o55 
.o54 
.o55 
.o54 
.o55 
.o55 
.o55 
.o55 
.o55 
.o55 
.o55 
.o56 
.o55 
.o55 
.o56 
.o56 
.o55 

Boiling  Water. 

J.ABLE     A.AV111.                                     OW 

Depression  of  Mercury  in  Glass  Tubes. 

Fah't      English 
Degrees.    Inches. 

Diff. 

Diameter 
of  Tube. 

Ivory. 

Young. 

Laplace. 

Poisson. 

Davendish. 

Daniell. 
Boiled 

33 

T 

uuetj. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch 

Inch. 

2OO  •  2 

.3 

.4 

.789 

.845 

.o56 
.o56 
.o56 

o.o5 
.10 

0.2949 

.i4o4 
.o865 

0.2964 
.1424 
.0880 

0. 
.i394 

.o854 

0.2796 
.l367 
.o83o 

O. 

.i4o 

.092 

0 

.070 

.044 

.6 

!s 

.901 

•957 
28.oi3 
.069 

.o56 
.o56 
.o56 
.  057 

.20 
.25 

.3o 

.35 

.o583 
.0409 
.0293 
.0212 

.0589 

.0280 
.0196 

.o58o 

.0412 

.0296 
.0216 

.o559 
.o394 
.0281 

.O2O4 

.067 

.o5o 
.o36 

.025 

.029 
.020 

.oi4 
.010 

•9 
209.0 
.  i 

.2 

.3 

.  126 

.182 
.239 
.295 
.352 

.o56 
.o57 
.o56 
.  057 

.4o 
.45 
.5o 
.60 

.oi54 
.0112 
.0082 
.oo43 

.OlSg 
.OIOO 
.0074 
.0045 

.oiSg 
.0117 
.0087 
.oo4<3 

.0109 
.0080 

.oo4i 

.oi5 
.010 
.007 
,oo5 

.007 
.oo5 
,oo3 

.002 

.  057 

.70 

.OO23 

.0024 

.0020 

.5 

!466 

.057 

0.80 

.0012 

.ooi3 

.0010 

.6 

.523 

.o57 

rj 

.58o 

.o57 

'  1 

.8 

.637 

.o57 
.o58 

•9 

.695 

.o57 

210.  O 

.752 

.o58 

.1 
.2 

.810 
.867 

.057 
.o58 

, 

.3 

.925 

.o58 

.4 

.983 

.o58 

.5 

29.041 

.o58 

.6 

.099 

.o58 

•7 
.8 

.i57 

.215 

.o58 
.  o5o 

•9 

.274 

.o58 

21  I  .0 

.332 

r 

.  I 

.39i 

.059 
.o58 

.2 

.449 

o5o 

.3 

.5o8 

.4 
.5 

.567 
.626 

•  OUQ 

.oSg 

.6 

•7 

'•744 

t'05o 

TABLE 

XXIX. 

.8 
•9 

212.0 

.8o3 
.863 
.922 

.982 

.  060 
.o59 
.060 

Factors  by  which  the  Difference  of  Readings  of  the  Dry-bulb  and  Wet- 
bulb  Thermometers  must  be  multiplied,  in  order  to  produce  the  Differ- 
ence between  the  Readings  of  the  Dry-bulb  and  Dew-point  Thermom- 

.2 

3o.o4i 

.059 

eters. 

* 

.3 

.101 

'060 

*% 

i! 
,0  •*-» 

*$ 

^fe 

ij 

*$ 
•S| 

•  4 

.161 

'06 

11 

3  g 

II 

l| 

.5 
.6 
.7 

.221 

.28l 
.341 

.060 
.060 

1 

1 
£ 

£>g 

1 

Sj 

1 
to 

i 

H 

1 

p| 

1 

jj 

1 

.8 

.4oi 

r* 

o 

0 

0 

o 

0 

0 

213.0 

.462 

.522 

.061 
.060 

/j 

20 

21 

8 

8 

.5 
.5 

32 

33 

2.8 

44 

45 

2.3 
2.3 

56 
57 

•9 
•9 

68 
69 

.6 

c 

80 

81 

.5 

.5 

.1 

.583 

.061 

22 

8 

.5 

34 

2.6 

46 

2.3 

58 

•9 

70 

c 

82 

.5 

.2 

.3 

.4 

.644 
.704 
.765 

.061 
.060 
.061 

23 

24 
25 

8 

6 

.5 
.3 
.4 

35 
36 
37 

2.6 
2.6 
2.5 

47 

48 

49 

2.2 
2.2 
2.2 

59 
60 

61 

.8 
.8 

71 

72 

73 

5 
5 

.'5 

83 

84 

85 

.5 
.5 
.5 

.5 

.826 

.061 

/j 

26 

6 

.  i 

38 

2.5 

5o 

2.  I 

62 

•7 

74 

.5 

86 

.5 

.6 

•7 

2l4-0 

.887 
•  948 
3  i  .009 
.071 

.132 

.061 
.061 
.061 
.062 
.061 

27 
28 
29 

3o 
3i 

6 
5 
5 
4 
3 

.  i 

•7 

.0 

.6 

•  7 

39 

4o 
4i 

42 

43 

2.5 
2.4 
2.4 

2.4 
2.4 

5i 

52 

53 
54 
55 

2.  I 
2.O 
2.0 
2.O 
2.0 

63 
64 
65 
66 
67 

•7 

.6 
.6 

75 
76 

77 
78 

79 

.5 
.5 
.5 
.5 
.5 

87 
88 
89 
90 

.5 
.5 
.5 
.5 

400 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation.               |  Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.       s. 

s. 

0            /               ,/ 

» 

I 

4 

21  Andromedae,       a 

i 

o     o  38.57* 

+  3.086 

61  44  16.1* 

—  I9.93 

2 

7 

1  1   Cassiopeae,         j3 

2^ 

I     12.  2O* 

3.i49 

3i  4o  39.9* 

19.89 

3 

ii 

Phcenicis,           e 

4 

I    47-34 

3.089 

i  36  34  24.7 

20.  O4 

4 

16 

22  Andromedae, 

5 

2    32.59 

3.093 

44  45  47.1 

20.  o5 

5 

J9 

Octantis,            y3 

5 

3     6.19 

2.820 

173     3  32.9 

19.89 

6 

26 

88  Pegasi,               Y 

2 

5  30.99* 

3.o84 

75  39     2.1* 

20.  04 

7 

52 

24  Andromedae,       0 

5 

9  16.17 

3.  in 

52     9     5.i 

20.  03 

8 

62 

8  Ceti,                   i 

4 

ii  47-o3* 

3.o6o 

99  39  22.0* 

19.98 

9 

64 

Tucanae,             <T 

5 

12    11.93 

3.i59 

i55  45  26.9 

21.  l4 

10 

70 

Tucanae,              vr 

4i 

i3  41.72 

2.834 

160  27  3i  .9 

19.88 

ii 

72 

Sculptoris,          t 

5 

i3  58.54 

3.025 

119  48  42.0* 

19.92 

12 

88 

Hydri,                /3 

3 

17  47-38 

3.297 

168     6     4-4 

20.26 

i3 

93 

Phoenicis,            K 

4 

18  49.23 

2.994 

i34  3o  5o.8 

19.79 

i4 

94 

Phoenicis,            a 

2 

18  5i.42 

2.983 

i33     7  11.7 

19.69 

i5 

io3 

Sculptoris, 

5 

20  29.10 

2.989 

123  5o     5.o 

2O,O2 

16 

121 

1  4  Cassiopese,         a 

5 

23  3i.35 

3.263 

36  18  22.7 

19.97 

ll 

124 

Phoenicis,           /I 

5 

24     9.91 

2.906 

i3g  38     2.8 

19.94 

18 

126 

1  5  Cassiopeae,         K 

4 

24  30.74* 

3.344 

27  53  48.2* 

19.96 

«f 

I27 

Tucanae,             ft1 

4 

24  38.29 

2.770 

i53  47     9.2 

I9.87 

20 

128 

Tucanae,             p* 

4 

24  39.13 

2.773 

i53  47  34.3 

I9.89 

21 

1  34 

Tucanae, 

5 

25  5i.93 

2.751 

i53  5i   36.8 

I9.5o 

22 

i43 

Phoenicis, 

5 

27  19.48 

2.901 

i43   12     5.6 

19.95 

23 

i53 

17  Cassiopeae,         C 

4 

28  38.45* 

3.299 

36  55  45.8* 

19.91 

24 

1  55 

29  Andromedae,       K 

41 

28  52.91 

3.i83 

57     6  25.8* 

19.92 

25 

1  64 

3o  Andromedae,       e 

4 

3o  38.35* 

3.i54 

61   3o  11.9* 

19.67 

26 

166 

3  1  Andromedae,       6 

3 

3i   18.99 

3.187 

59  57  38.5* 

19.76 

27 

169 

1  8  Cassiopeae,         a 

3 

32     1.58* 

3.354 

34  17     9-9* 

19.83 

28 

176 

Tucanae, 

5 

33  21.60 

2.8o5 

i5o  18     1.6 

20.  19 

29 

i83 

Phoenicis,           {* 

5 

34  i3.89 

2.869 

i36  54  27.5 

20.  03 

3o 

188 

Phoenicis,            £ 

5 

34  54.  5i 

2.738 

i47   19  33.9 

19.96 

3i 

189 

20  Cassiopeae,         TT 

5 

35  ii.45 

3.289 

43  4?  47.9 

I9.8l 

32 

192 

Sculptoris,          A1 

5 

35  29.35 

2.901 

129  17     8.4 

19.89 

33 

196 

1  6  Ceti,                   (3 

2i 

36     3.42* 

3.oi6 

108  48  39.i* 

19.86 

34 

199 

Phoenicis,           77 

5 

36  35.64 

2.713 

i48  17  17.3 

19.29 

35 

200 

17  Ceti,                   0l 

5 

36  37.45 

3.o3i 

101   25  37.1 

I9.70 

36 

202 

Sculptoris,          P 

5 

36  57.o3 

2.919 

129  i4  56.6 

19.88 

37 

2l5 

34  Andromedae,       £ 

4 

39  23.79* 

3.169 

66  32  59.4* 

19.69 

38 

2!8 

24  Cassiopeae,         TJ 

4 

4o     3.4o* 

3.564 

32  58  53.5* 

19.27 

39 

2I9 

25  Cassiopeae,         o> 

5 

4o  21  .66 

3.359 

39  5i     7.9 

19.68 

4o 

222 

63  Piscium,             6 

5 

4o  54.20* 

3.107 

83  i3  56.5* 

I9.73 

4i 

227 

35  Andromedae,       v 

4 

4i   33.42* 

3.279 

49  44  19.9* 

19.72 

42 

242 

20  Ceti, 

5 

45  20.  57* 

3.o64 

91  57  35.4* 

19.67 

43 

245 

Cassiopeae, 

5 

46  35.01 

3.369 

42     8     7.8 

19.64 

44 

253 

27  Cassiopeae,         y 

3 

47  4i.83* 

3.548 

3o     5  48.i* 

19.64 

45 

25g 

37  Andromedae,      // 

4 

48  26.48* 

3.3oi 

52  18  54.o* 

19.68 

46 

262 

2  Ursae  Minoris, 

5 

49    9-55* 

6.716 

4  33     2.4* 

19.59 

47 

264 

38  Andromedae,       rj 

5 

49  i2.5o 

3.  190 

67    23    32.1 

19.60 

48 

272 

Sculptoris,          a 

5 

5l     22.  5l* 

2.899 

120    10       8.7*         19.52 

49 

288 

7  1    Piscium,              e 

4 

55     9.75* 

3.n4 

82  55     7.2* 

ig.So 

5o 

3i7 

Phoenicis,           /? 

3* 

o  5g  23.  10 

+2.692 

137  3i   21.3 

-I9-37 

TABLE    XXX. 


401 


CATALOGUE   OF    1500    STARS. 


No. 

Logarithms  of                                 j|                                  Logarithms  of 

a 

b                  c 

d 

a' 

b'               c' 

d' 

I 

+8.8790 

-f6.3273 

+  o.4875 

4-8.5544 

—9.58o8 

-9.6753 

—     .3O22 

+7-4483 

2 

9.  io36 

6.823o 

0.4888 

+9.o336 

9.3491 

—9.9299 

.3O22 

7.7194 

3 

8.9867 

6.8791 

o.4857 

—8.8478 

9.4829 

+9.8611 

.3O22 

7.8924 

4 

8.9762 

7.0214 

0.4893 

+8.8275 

9.4738 

—  9.85i2 

.3022 

8.o452 

5 

9.7416 

7.8733 

0.4657 

-9.7385 

8.8182 

+9.9968 

.3O22 

8.i3!6 

6 

8.8375 

7.2192 

0.4884 

+8.23i7 

9.6174 

—  9.394o 

.3021 

8.38i5 

7 

8.9261 

7.5333 

o.493i 

+8.7i4o 

9.5o23 

-9.7876 

.3OI9 

8.6068 

8 

8.8295 

7.54ii 

o.4856 

—  8.o542 

9-6399 

+9.2241 

.3017 

8.7110 

9 

9.2099 

7.9364 

0.4643 

—  9.  1698 

9.3555 

+9-9593 

.3oi6 

8.7269 

10 

9.2988 

8.o757 

0.4542 

—  9.2730 

9.3o43 

+9.9735 

.3oi4 

8.7761 

ii 

8.8848 

7.67o5 

0.4806 

—  8.58i2 

9.6094 

+9.6957 

.3oi4 

8.7849 

12 

9.5o83 

8.3982 

o.4n6 

—9.4989 

9.2180 

+9.9893 

.3009 

8.8886 

i3 

8.9693 

7.8847 

0.4717 

—  8.8i5i 

9.5647 

+9-8443 

.3oo8 

8.9140 

i4 

8.9592 

7.8754 

0.4724 

-8.7939 

9.5717 

+9.8333 

.3oo8 

8.9i48 

i5 

8.9028 

7.8552 

o.4758 

—8.6485 

9-6i3o 

+9.7440 

.3oo5 

8.9607 

16 

9.0492 

8.0620 

o.5i28 

+8.9555 

9.2416 

—  9.9040 

.2999 

9.0106 

*7 

9.0101 

8.o348 

0.4632 

—  8.892i 

9.5579 

+9.8795 

.2998 

9.0222 

18 

9.i5i3 

8.1822 

0.5238 

+9-°976 

9.o358 

—9.9439 

.2997 

9.0284 

'9 

9.1763 

8.2094 

0.4439 

—  9.i29i 

9.4594 

+9.9504 

.2997 

9.o3o6 

20 

9.i763 

8.2097 

0.4439 

—  9.  I292 

9.4592 

+9.9504 

.2997 

9.o3o8 

21 

9.1771 

8.23i5 

o.44i5 

—  9.i3o3 

9.4658 

+9.9504 

.2995 

9.0616 

22 

9.o434 

8.1218 

o.456i 

—8.9469 

9.55o4 

+9.9004 

.2991 

9.0763 

23 

9.04  1  8 

8.i4o8 

o.5i75 

+8.9445 

9.2071 

—9.8994 

.2988 

9.0967 

24 

8.8963 

7.9991 

o.5o23 

+8.63i2 

9.4714 

—9.73i4 

.2988 

9.0993 

25 

8.8761 

8.oo48 

0.5007 

+8.5547 

9.5O22 

-9.6747 

.2983 

9.1248 

26 

8.8825 

8.0208 

0.5019 

+8.5820 

9.4879 

—9.6954 

.2982 

9.1342 

27 

9.0689 

8.2170 

0.5242 

+8.9860 

9.1119 

—  9.9I29 

.2980 

9.i439 

28 

9.1243 

8.2904 

0.4363 

—  9.o63i 

9.5328 

+  9.9342 

.2976 

9.  1616 

29 

8.9845 

8.1620 

o.456i 

—8.848o 

9.6077 

+9.  8586 

.2974 

9.1726 

3o 

9.0866 

8.2727 

o.44oo 

—  9.0118 

9.5587 

+9.9201 

.2972 

9.  1811 

3! 

8.9786 

8.i683 

o.5i64 

+8.837o 

9.2785 

—  9.8533 

.2971 

9.1845 

32 

8.9299 

8.1233 

0.4627 

—  8.73i5 

9.637i 

+9.7963 

.2970 

9.1882 

33 

8.8424 

8.0428 

0.4770 

—  8.35o8 

9.664o 

+9.5o3i 

.2968 

9.  1960 

34 

9.0977 

8.3o46 

0.4356 

—  9.0274 

9.56o4 

+9.9242 

.2967 

9.20l4 

35 

8.8270 

8.o343 

o.48n 

—  8.1240 

9.6598 

+9.2914 

.2967 

9.2OI7 

36 

8.9293 

8.i4o5 

0.4617 

—  8.73o5 

9.6412 

+9.7955 

.2966 

9.2066 

37 

8.8549 

8.0945 

o.Soio 

+8.4547 

9.5i85 

—  9.5934 

.2958 

9.2332 

38 

9.0814 

8.3283 

o.535i 

+9.0060 

8.9557 

—9.9170 

.2956 

9.24O2 

39 

9  .  o  i  o4 

8.2607 

O.5252 

+8.8956 

9.  1572 

—9.8784 

.2955 

9.2435 

4o 

8.8200 

8.0762 

0.4912 

+  7.8914 

9.6128 

—  9.0644 

.2953 

9.2493 

4i 

8.9341 

8.1974 

0.6162 

+8.  7446 

9.33i6 

—9.8032 

.295© 

9.2661 

42 

8.8:56 

8..  1  1  77 

o.486o 

—7.3496 

9.6439 

+8.5254 

.2937 

9.2935 

43 

8.9882 

8.3o23 

0.5275 

+8  ..8584 

9.i5o5 

—9.8611 

.20,32 

9.3o5i 

44 

9.1142 

8.4388 

0.5499 

+9.061  3 

8.6900 

-9.9276 

.2928 

9.3i5i 

45 

8.9I57 

8.2474 

o..5i68 

+8-7020 

—9.3326 

-9.7766 

.2924 

9.3218 

46 

9.9143 

9.2523 

0.8224 

+9.9i3o 

+9.2497 

—9.9886 

.2922 

9.3279 

4? 

8.8485 

8.1872 

o.5o37 

+8.4334 

—  9.5o34 

—9.5747 

.2921 

9.3286 

48 

8.8761 

8.234i 

0.4621 

-8.5773 

9.6874 

+9.6902 

.2912 

9.3469 

49 

8.8i45 

8.2o45 

0.4928 

+7.9o54 

9.6o35 

—9.0782 

.2895 

9-3773 

5o 

+8.9797 

+8.4o3o 

+o.43o8 

—8.8475 

—  9.683i 

+9.853o 

—     .2875 

+9.4086 

Cc 


402 


TABLE   XXX. 


CATALOGUE   OF    1500    STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.. 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.         s. 

s. 

O             /                 // 

5l 

3i8 

4  1  Andromedse, 

5 

o  59  25.  3o 

+  3.4IO 

46  5i   3o.i 

—  I9.35 

52 

328 

80  Piscium,             e 

5 

i     o  38.79* 

3.o83 

85     8  4i.8* 

I9.i7 

53 

33o 

4a  Andromedae,       <j> 

5 

o  49-n* 

3.437 

43  33  34.i* 

19.37 

54 

332 

3  1  Ceti,                   jj 

31 

i     2.67 

3.019 

loo  58  42.5 

19.23 

55 

333 

Tucanae,             i 

5 

i   19.96 

2.357 

162  34  36.  9 

19.49 

56 

334 

43  Andromedae,       j3 

2 

i  20.90* 

3.336 

55   10  33.2* 

19.27 

5? 

339 

33  Cassiopeae,          6 

4* 

i  59.95* 

3.593 

35  38  58.3* 

19.33 

58 

34o 

Phcenicis,            £ 

5 

2     4.35 

2.555 

i46     3     i.5 

i8.94 

59 

348 

84  Piscium,             x 

5 

3  24.08 

3.2IO 

69  45  52.6 

io/.  28 

60 

36o 

I  Ursae  Minoris,    a 

2 

5     o.83* 

17.546 

i   29  24.7* 

19.27 

61 

38o 

Phcenicis,            v 

44 

8  25.  i5 

2.725 

i36  19  59.4 

I9.53 

62 

392 

Tucanae,             K 

5 

10  4i  .92 

2.089 

i  5g  4o  29.8 

I9.  IO 

63 

398 

Tucanae, 

5 

ii   5o.37 

2.117 

157   ii    3i.3 

18.84 

64 

4o4 

46  Andromedae,       f 

41 

i3  3i.85 

3.493 

45  'i5  34.4 

I9.00 

65 

4l2 

36  Cassiopeae,         i/> 

41 

i5  24.16* 

4.119 

22  39   18.  i* 

I9.OI 

66 

4i6 

37  Cassiopeae,          d 

3 

16     2.88* 

3.853 

3o  32  47.3* 

18.92 

67 

420 

45  Ceti,                   0 

3 

16  3i.58* 

3.000 

98    57     32.2* 

18.76 

68 

422 

Tucanae, 

5 

16  47.59 

2.o4l 

i57  10   19.1 

18.62 

69 

426 

Phoenicis, 

5 

l8       2.32 

2.666 

i32   16  26.8 

18.94 

70 

427 

93  Piscium,             p 

5 

18  10.  60 

3.  216 

71    36  34.5 

18.96 

7* 

429 

46  Ceti, 

5 

18   14.70 

2.952 

io5  22   5o.6 

18.91 

.72 

43i 

g4  Piscium, 

5 

18  36.  06 

3.225 

71    32   16.7 

18.88 

73 

432 

48  Andromedae,       u 

5 

18  42.29* 

3.55o 

45    22     10.6* 

18.78 

74 

.438 

38  Cassiopeae,          A 

5 

20     9.01 

4.324 

20  3o  36.  o 

18.76 

75 

44i 

49  Andromedae,       A 

5 

21      7.73 

3.556 

43  46     5.5 

18.78 

76 

447 

Phcenicis,            y 

3 

21   5i.i6 

2.634 

i  34     5   i4.8 

i8.64 

77 

448 

98  Piscium,             p 

41 

22     19.  73* 

3.!37 

84  37  53.2* 

18.60 

78 

453 

99  Piscium,             TJ 

4 

23  27.80* 

3.  200 

75  25   45.o* 

18.76 

79 

46  1 

Phoenicis,           J 

4 

25       O.II 

2.509 

i39   5i    i5.6 

iS-77 

80 

48o 

5o  Andromedae, 

5 

28     0.81* 

3.489 

49  20  48.7* 

18.22 

II 

487 

5  1  Andromedae, 

31 

28  48.69* 

3.639 

42     8     2.5* 

i8.45 

82 

488 

1  02  Piscium,              TT 

5 

29     9.21* 

3.171 

78   37  38.i* 

18.64 

83 

502 

53  Andromedae,       T 

5 

3i  44.87 

3.5i6 

5o  ii     8.1 

i8.38 

84 

5o7 

Eridani,              a 

i 

32     7.23 

2.238 

i48     o     o.5 

i8.45 

85 

5i8 

1  06  Piscium,             v 

5 

33  37.69* 

3.117 

85    16  25.3* 

i8.38 

86 

522 

54  Andromedae, 

4 

34  17.28* 

3.716 

4o    4     9.1* 

18.35 

87 

536 

52  Ceti,                   T 

31 

37     6.09 

2.788 

i  06  43  43.5 

19.  i5 

88 

537 

no  Piscium,             o 

5 

37  28.67* 

3.162 

81   35   56.  9* 

i8.3i 

89 

54i 

Sculptoris,          £ 

5 

38  37.35 

2.8l9 

ii5  48   14.9* 

18.24 

90 

55o 

Eridani,              g* 

5 

4o  23.  3o 

2.287 

i  44  16  37.6 

17.97 

91 

559 

53  Ceti,                    x 

5 

42   i  3.  oo* 

2.946 

101   z5  49.9* 

18.00 

92 

564 

45  Cassiopeae,          e 

3 

43  39.68* 

4.221 

27    4  17.9* 

18.04 

93 

565 

55  Ceti,                   C 

3 

44     3.54 

2.960 

101     4  42.9 

1-7.90 

94 

569 

2  Trianguli,           a 

31 

44  32.  57* 

3.399 

61     9  i4-9* 

17.80 

95 

572 

5  Arietis,               y1 

41 

45  i8.43* 

3.277 

71   26  38.5* 

17.89 

96 

573 

5  Arietis,               yz 

41 

45  i8.43* 

3.279 

71   26   29.0* 

17.88 

97 

577 

6  Arietis,               ft 

3 

46  21  .69* 

3.297 

69  55   38.  7* 

17.82 

98 

582 

Phcenicis, 

5 

47  37.92 

2.4l2 

l37       2     17.7 

17.79 

99 

585 

Phoenicis,            0 

5 

48    8.43 

2.49I 

i33   i4     1.8 

17.90 

IOO 

589 

Hydri,                 r/1 

5 

i  48  45.68 

+  I.458 

168  4o  58.3 

—  17.96 

TABLE   XXX. 


403 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a                  b                 c 

d 

of  .                i/ 

c' 

d' 

5i 

+8.9460 

+8.3697 

+o.53o4 

+8.7810 

—  9.  i5oi 

—  9.8202 

—    .2875 

+9.4089 

52 

8.8101 

8.243o 

0.4914 

+  7.7376 

9.6i3i 

—  8.9121 

.2868 

9.4i75 

53 

8.9702 

8.4o44 

0.5365 

+8.83o3 

9.o374 

—9.8447 

.2868 

9-4i87 

54 

8.8i63 

8.2523 

0.4775 

—8.0961 

9.6777 

+9.2642 

.2866 

9.42o3 

55 

9.1449 

8.5829 

o.3783 

—  9.0931 

9.638i 

+9.9325 

.2865 

9.4223 

56 

8.8939 

8.3320 

O.52O7 

+8.65o6 

9.3l22 

—9-7409 

.2865 

9.4224 

57 

9.0423 

8.4852 

o.5525 

+8.9522 

8.5539 

—9.8938 

.2861 

9.4269 

58 

9.0608 

8.5o43 

o.4o47 

—  8.9796 

9.6668 

+9-9027 

.2861 

9.42-74 

59 

8.8348 

8.2880 

o.SoSg 

+8.3737 

—  9.495i 

—  9.  5221 

.2854 

9.4364 

60 

o.Sgii 

9.8559 

i  .2420 

+0.3909 

+9.4a89 

—  9.9821 

.2845 

9.4470 

61 

8.965i 

8.4533 

0.424? 

—8.8245 

—  9.7o96 

+9.8397 

.2826 

9.  4685 

62 

9.2622 

8.7655 

0.2955 

—  9.2342 

9.6389 

+9.95n 

.2812 

9.4823 

63 

9.2i38 

8.7245 

O.32O2 

—  9.i784 

9.6555 

+9'9429 

.2805 

9.4891 

64 

8.9497 

8.47i3 

O.5426 

+8.7972 

—  8.9355 

—  9.8248 

.2795 

9.4988 

65 

9.2143 

8.7476 

o.6i34 

+9.i794 

+9.n73 

—9.9412 

.2783 

9.5094 

66 

9.0935 

8.63o7 

o.58o8 

+9.o286 

+8.7774 

—9.9108 

.2779 

9.5129 

67 

8.8o46 

8.3448 

0.4773 

—  7«9969 

—  9.68io 

+9«l677 

.2775 

9.5i55 

68 

9.2103 

8.752i 

o.3o67 

—9.1749 

9.6735 

+9.9397 

.2774 

9.5i7o 

69 

8.9290 

8.4784 

o.4257 

-8.7569 

9.737o 

+9.8021 

.2765 

9.5237 

7° 

8.8209 

8.3711 

o.5o78 

+8.3i99 

9.4862 

—9.4732 

.2765 

9.5244 

?i 

8.8139 

8.3646 

0.4695 

—  8.2376 

9.7o52 

+9.3978 

.2764 

9.5248 

72 

8.8208 

8.3736 

o.5o8o 

+8.32i4 

9-4844 

—9.4746 

.2-762 

9.526-7 

73 

8.9455 

8.4989 

0.5459 

+  8.7922 

-8.859i 

—9.8205 

.276l 

9.5273 

74 

9.2523 

8.8142 

o.633o 

+  9-2238 

+9.22-76 

—9.9444 

.275l 

9.5348 

75 

8.9562 

8.5239 

o.55o8 

+8.8i48 

—8.6981 

—  9.83o8 

.2744 

9.5399 

76 

8.9393 

8.5ii2 

0.4179 

—  8.78i8 

9  .  -7442 

+9.8142 

.2739 

9.5436 

77 

8.7972 

8.37i8 

0.4934 

+7.7682 

9.6012 

—8.9424 

.2736 

9.546o 

78 

8.8087 

8.3898 

o.5o44 

+8.2093 

9.5190 

—9.3712 

.2728 

9.55i7 

79 

8.9840 

8.5738 

0.39-72 

—  8.8673 

9.7458 

+9.8528 

.2716 

9.5592 

80 

8.9110 

8.5i75 

0.5442 

+8.7249 

8.9299 

-9.7811 

.2694 

9.5736 

81 

8.9638 

8.5746 

o.5598 

+8.834o 

8.o374 

—  9.8367 

.2688 

9.5774 

82 

8.7988 

8.4n5 

o.5oi4 

+8.0937 

9.5448 

—  9.2612 

.2685 

9.579o 

83 

8.9027 

8.5292 

0.5447 

+8.7o9i 

8.9232 

—  9.77o6 

.2665 

9.59o8 

84 

9.o636 

8.6921 

0.3490 

—  8.9921 

9.7496 

+9.8924 

..2662 

9.5924 

85 

8.7881 

8.4245 

0.4934 

+7.7040 

—9.6018 

—  8.8786 

.2649 

9.599i 

86 

8.9774 

8.6172 

o.5689 

+8.8612 

+8.4249 

—9.8460 

.2644 

9.6020 

87 

8.8025 

8.4567 

O.4632 

—8.2617 

—9.7275 

+9.4190 

.2620 

9.6140 

88 

8.7881 

8.4442 

0.4986 

+7.9527 

9.567o 

—9.1241 

.2617 

9.6i56 

89 

8.8280 

8.4898 

0.4473 

—8.4668 

9.7576 

+9.5973 

.2607 

9.62o3 

9° 

9.0145 

8.685i 

0.3583 

—  8.9240 

9.7766 

+9.8664 

.2592 

9.6275 

91 

8.7879 

8.4675 

0.4704 

—  8.o85o 

—  9.7o83 

+9.2523 

.2575 

9.6349 

92 

9.  1198 

8.8o63 

o.6247 

+9.0694 

+9.2824 

—  9.9036 

.2562 

9.64o5 

93 

8.7857 

8.4742 

0.470-7 

—8.0693 

—9.7077 

+9.2373 

.2558 

9.6421 

94 

8.8346 

8.5254 

o.53o8 

+8.5i8i 

9.2243 

—  9.6366 

.2554 

9.644o 

95 

8.7995 

8.4940 

o.5i45 

+8.3023 

9.43i9 

—9.4562 

.2547 

9.6469 

96 

8.7995 

8.4940 

o.5i45 

+8.3o23 

9.43i7 

—9.4552 

.2547 

9.6469 

97 

8.8026 

8  .  5o2o 

o.5i7i 

+8.338i 

9.4048 

—9.48-70 

.2537 

9.6509 

98 

8.94o7 

8.646i 

o.384o 

—  8.8o5i 

9.79-72 

+9.8i46 

.2626 

9.6557 

99 

8.9112 

8.6190 

o.3978 

—8.  -7469 

9-7977 

+9.7854 

.2520 

9.6576 

IOO 

+9.2125 

+8.9232 

+o.i775 

—9.181-7 

—  9.  -7660 

+9.9184 

-  .25i4 

+9.6599 

404 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.   m.         s. 

s. 

o         /            ~,  

101 

595 

48  Cassiopese, 

5 

I    49   43.96* 

+4.778 

i9  49  26.5* 

—  17.79 

I  O2 

596 

Eridani,             % 

4 

5o     6.71 

2.325 

l42     21     25.7 

18.09 

io3 

600 

5o  Cassiopeae, 

4 

5o  44.28* 

4.940 

i  8   i  8  28.9* 

17.78 

io4 

6o3 

Hydri,                ^ 

4} 

5i     7.99 

1,486 

i58  a3   12.9 

17.75 

io5 

618 

5  9  Ceti,                  v 

4 

52    56.22 

2.827 

in  48  22.7 

17.69 

1  06 

623 

Hydri,                  a 

3 

54     2.62 

1.889 

i52  18     4.7 

17.62 

107 

625 

u3  Piscium,             a 

3* 

54  17.50 

3.102 

87  57  44-7 

17.62 

1  08 

628 

57  Andromedar,       7 

3 

54  42.71* 

3.644 

48  23  33.9* 

17.56 

109 

634 

PhcEnicis,           x 

5 

55  4i.2i 

2.4O2 

i35  26  ii.  5 

17.79 

no 

635 

Hydri, 

5 

55  43.87 

I  .570 

i56  47  55.7 

17.01 

III 

648 

1  3  Arietis,               a 

2 

58  43.64* 

3.364 

67  14  57.9* 

I7.3o 

112 

656 

4  Trianguli,          (3 

4 

2       0    38.02* 

3.545 

55  43  3o.o* 

17.32 

ii3 

684 

65  Ceti,                   £l 

5 

5     3.28* 

3.169 

81   5i   34.o* 

17.12 

n4 

688 

Fornacis,            [t 

5 

6  17.93 

2.642 

121     25    47.3* 

17.01 

ii5 

717 

Eridani,             0 

4 

ii     9.34 

2.i56 

l42     12    26.7 

16.91 

116 

720 

68  Ceti,                   o 

Var. 

ii  46.53 

3.021 

93  39  4i.8 

16.60 

117 

721 

9  Persei,                i 

5 

ii  56.19* 

4.n4 

34  5o  39.6* 

16.82 

118 

744 

Cassiopeae,, 

4 

16  46.71* 

4.812 

2.3  16  34.3* 

i6.59 

119 

?54 

72  Ceti,                   p 

5 

18  42.42 

2.897 

102  58     9.6 

16.47 

120 

756 

Hydri,                6 

4 

19     5.79 

1.037 

i59  20  36.3 

i6,44 

121 

760 

73  Ceti,                   s£2 

4 

20    11.34* 

3.182 

82     12    54.0* 

16.42 

122 

763 

Eridani,             K 

4| 

21     28.98 

2.186 

i38  22  48.i 

15.99 

123 

781 

76  Ceti,                   a 

5 

24  58.  93 

2.844 

io5  54  i9.4 

16.  10 

124 

794 

78  Ceti,                   v 

4* 

28     0.42* 

3.i35 

85     3  5o.7* 

15.99 

125 

811 

82  Ceti,                   6 

4 

3i   48.i3 

3.o74 

9o  i9  i8.5 

i5.78 

126 

8i5 

83  Ceti,                   e 

4 

32   18.69 

2.899 

102  3o  4i  -o 

i5.56 

127 

827 

1  3  Persei,                6 

4 

33  58.  81* 

4.o5i 

4i   24  35.o* 

16.60 

128 

828 

Eridani, 

5 

34     4-65 

2.276 

i33  32   i5.i 

i5.65 

129 

83i 

35  Arietis, 

4 

34  39.69 

3.499 

62  56     2.1 

i5.68 

i3o 

832 

Eridani,              t 

4 

34  45.10 

2.378 

i3o  29  56.2 

16.66 

i3i 

837 

86  Ceti,                   7 

3 

35  3i.93* 

3.102 

87  23  57.8* 

i5.45 

I  32 

845 

Arietis, 

4 

36  5o.37* 

3.232 

80  3i   20.5* 

i5.49 

i33 

847 

89  Ceti,                   TT 

4 

36  59.13 

2.85i 

io4  29  49«i 

i5.5i 

1  34 

849 

Hydri,                e 

5 

37  17.66 

0.861 

i  58  54  42.  9 

15.29 

i35 

856 

i   Eridani,              rr 

4 

38     6.34 

2.800 

109  12  37.o 

i5.5o 

i36 

86  1 

Sg  Arietis, 

4 

38  Sg.So* 

3.549 

61  22  44-5* 

i5.3i 

i37 

863 

1  5  Persei,                77 

4 

39  47-  5o* 

4.312 

34  43  52.4* 

i5.38 

1  38 

870 

42  Arietis,                TT 

5 

4o  55.74* 

3.335 

73     9  45.5* 

i5.35 

i39 

871 

1  6  Persei, 

4* 

4i     7.91 

3.759 

52  18     9.3 

l5.22 

i4o 

872 

4  1  Arietis, 

3 

4i     9.98* 

3.5io 

63  21  40.7* 

1  5.  20 

i4i 

879 

Fornacis,           /? 

5 

42  48.68 

2.509 

123       2     17.9* 

i5.4i 

142 

882 

Hydri,     '            C 

5 

43  14.78 

o.874 

i58   i4  54.o 

i5.i3 

i43 

885 

1  8  Persei,                r 

5 

43  39.o5* 

4.199 

37  5i   20.3* 

i5.i5 

i44 

887 

2  Eridani,              r2 

4* 

44  i4.i5 

2.720 

in   37  28.9 

i5.ii 

i45 

910 

3  Eridani,             i) 

3 

49    6.07 

2.928 

99  29  53.o 

14.62 

1  46 

912 

22  Persei,                TT 

5 

49    II  .22* 

3.8o6 

5o  56  29.o* 

i4.8o 

i47 

921 

48  Arietis,               e 

5 

5o  38.  60* 

3.4i8 

69  i5  46.4* 

i4.75 

1  48 

93i 

Horologii, 

5 

5i  43.49 

i  .226 

i53  43  28.3 

i4.68 

1  49 

937 

Eridani,              6 

3^ 

52    34-47 

2.269 

i3o  54  3o.5 

i4.63 

i5o 

947 

23  Persei,                7 

3i 

2  53  57.70* 

+4.296 

37     5     8.1* 

—  i4.55 

TABLE   XXX. 


405 


CATALOGUE   OF    1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

g 

c' 

d' 

101 

+9-24I7 

+8.957o 

+0.6794 

+  9-2l52 

+9.4567 

-9.9217 

—  i.25o4 

+9.6634 

IO2 

8.9859 

8.7029 

o.3559 

-8.8845 

—  9.8ooo 

+9.8465 

i.25oo 

9.6648 

io3 

9.2740 

8.9939 

o  .  694^ 

+9,25i5 

+9-4840 

—  9-9247 

1.2494 

9.667-1 

104 

9.2O45 

8.9262 

o.  1754 

—  9.i728 

-9.7732 

+9.9151 

1.2490 

9.6685 

106 

8.  Son 

8.53n 

0.4499 

—  8.37n 

—9.  -7624 

+9-5i49 

1.2472 

9.6749 

1  06 

9.  ioo5 

8.8354 

0.2683 

—  9.0476 

—9.  -7954 

+9.8910 

1.2461 

9.6788 

107 

8.7678 

8.5o39 

0.4904 

+  7.3i87 

—  9.6198 

—8.4945 

1.2458 

9.6797 

io3 

8.8934 

8.63i3 

o.56i2 

+8.7i55 

—  7-77°9 

—  9-7654 

1.245^ 

9.6811 

109 

8.9109 

8.6623 

0.3828 

-8.7727 

—  9.8i23 

+9-7949 

1  .  244^ 

9.6845 

I  10 

9.  1706 

8.9131 

o.i936 

—9.1339 

—9-  7894 

+9.9065 

1.2443 

9.6847 

ii  i 

8.7980 

8.5539 

0.5248 

+8.3854 

-9.3i95 

—  9.5263 

I  .2411 

9.6948 

I  12 

8.8436 

8.6078 

0.5476 

+8.5942 

-8.879i 

-9.6875 

I.239I 

9.7011 

n3 

8.  7602 

8.5435 

o.5on 

+  7.9II2 

-9.55i6 

—  9.o829 

I.234I 

9.7162 

n4 

8.8232 

8.6118 

0.4221 

—  8.54o5 

—9.8084 

+9-6476 

1.2326 

9.7190 

n5 

8.9612 

8.7702 

0.3298 

—8.8590 

—  9.84i3 

+9.8224 

1.2269 

9.7336 

116 

8.7487 

8.56o3 

o.48o6 

-7.5539 

—  9.67oi 

+8.7291 

I  .2261 

9.7355 

117 

8.9907 

8.8029 

o.6i45 

+8.9049 

+9.2986 

-9.8379 

I  .2259 

9.7359 

118 

9.1447 

8.9769 

0.6828 

+9.io78 

+9.5376 

—  9.88o7 

I.2I9S 

9-7497 

119 

8.7502 

8.5902 

0.4617 

—8.10x3 

—  9.74o8 

+  9.2662 

I.  2173 

9.755i 

120 

9.  1910 

9.0326 

O.O2O4 

—  9.  1621 

-9.8368 

+9.8857 

I  .2168 

9-756i 

121 

8.7410 

8.5870 

o.5oi8 

+  7.8?28 

—9.5468 

—  9.o449 

i.2i53 

9.7591 

122 

8.9130 

8.7642 

0.3423 

—8.7867 

-9.8579 

+9.785o 

i.2i36 

9.  7626 

123 

8.7475 

8.6126 

o.454i 

—8.1853 

—  9.763o 

+9-3444 

I.  2088 

9-77i8 

124 

8.7279 

8.6o5o 

0.4969 

+  7.6625 

—  9.58o5 

—  8.837o 

i  .2046 

9-7795 

125 

8.7207 

8.6128 

0.4866 

—  6.47o4 

—  9.6410 

+  7.6465 

1.1990 

9.7889 

126 

8.73o4 

8.6244 

o.46o5 

—8.0662 

—  9.746i 

+9.23i8 

i.i983 

9.  -790  1 

127 

8.8970 

8.7975 

o.6o36 

+8.7721 

+9.2548 

—  9.7686 

i.i958 

9-794i 

128 

8.857i 

8.7580 

o.3578 

—  8.6952 

—9.8-713 

+9.73!5 

i.i957 

9-7943 

129 

8.7668 

8.6700 

0.5438 

+8.4249 

—  9.oo9o 

—  9.55o6 

1.1948 

9-7957 

i3o 

8.8353 

8.7388 

o.3723 

—8.6478 

—9.8668 

+9-7°49 

1.1946 

9-7959 

i3i 

8.7r56 

8.6221 

0.4926 

+7.3724 

—  9.6o75 

—8.548i 

1.1934 

9-7978 

132 

8.7191 

8.6307 

0.5067 

+7.9357 

—  9.5io4 

—  9.io58 

1.1914 

9.8008 

i33 

8.7269 

8.639i 

o.455i 

—8.1255 

—  9.7623 

+9.2875 

1.1912 

9.8012 

134 

9.i564 

9.0697 

9«94i3 

—  9.1262 

-9.8735 

+9.8584 

1.1907 

9.8010 

i35 

8.736o 

8.6525 

o.443i 

—8.2533 

—9,7918 

+9-4o45 

i.i895 

9.8o38 

i36 

8.7664 

8.6863 

o.5487 

+8.4467 

-8.8727 

—9.5662 

1.1881 

9.8o58 

i37 

8.9528 

8.8758 

o.6344 

+8.8676 

+9-4479 

—9-7994 

i.  1868 

9.8076 

i38 

8.7257 

8.653i 

0.5227 

+8.i876 

—9.3585 

—9-3447 

i.i85o 

9.8lO2 

139 

8.8080 

8.7362 

0.5-727 

+8.5944 

+8.7i6o 

—9.6688 

1.1847 

9.8106 

i4o 

8.755o 

8.6833 

0.5446 

+8.4067 

—8.9912 

—9.5340 

i.i846 

9.8107 

i4i 

8.7802 

8.7i49 

0.3986 

—  8.5i68 

—  9.857o 

+9.6163 

1.1820 

9.8143 

142 

9.i34o 

9.0703 

9.9447 

—  9.  IO2O 

—9.  8853 

+9.8469 

1.  1812 

9.8i53 

143 

8.9143 

8.8522 

o.623o 

+  8.8117 

+9.3993 

-9-7757 

i.  1806 

9.8162 

i44 

8.7330 

8.6731 

o.435o 

—  8.2995 

—  9,8o99 

+9.4439 

1.1796 

9.8i75 

i45 

8.6991 

8.6577 

0.4654 

—  -7.9166 

—  9.73i6 

+9.0867 

i.i7i4 

9.8278 

1  46 

8.8028 

8.7618 

0.5-799 

+8.6022 

+8.9395 

—  9.6685 

i.i7i3 

9.8280 

m 

8.7i95 

8.684o 

0.5332 

+8.2686 

-9.2199 

—  9.4i56 

1.1687 

9.83io 

i48 

9.0424 

9.0110 

o.o885 

—  8.995i 

—9.9046 

+9.8i73 

1.1668 

9.  833s 

149 

8.8086 

8.78o5 

o.3576 

—8.6248 

—9.8898 

+9-6793 

i.i653 

9.835o 

i5o 

+8.9042 

+8.88i3 

+o.6323 

+8.8o6t 

-'-<).  4583!—  9.7625 

—  1.1629 

+9-8377 

406 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.       s. 

s. 

0            t               „ 

l5l 

948 

Persei, 

5 

2  54  18.40* 

+4.44o 

33  53   16.6* 

—  l4.6l 

162 

949 

92  Ceti,                  a 

8i 

54  26.60* 

3.  I29 

86  3o     7.4* 

14.42 

i53 

952 

9  Eridani,             p3 

5 

55  20.58 

2.o37 

98  16  42.0 

14-48 

1  54 

953 

25  Persei,               p 

4 

55  34.8i 

3.8i5 

5i  44  4o.5 

14.37 

i55 

954 

ii  Eridani,              r3 

4 

55  46.82 

2.643 

n4  12  56.i 

i4.36 

166 

956 

Horologii, 

5 

55  55.87 

I  .  IO9 

i  54  4o    7.2 

i4.43 

167 

959 

10  Eridani,             p3 

5 

56  54.63 

2.943 

98  ii  26.6 

i4.37 

1  58 

962 

Persei,                t 

4 

58  i5.67* 

4.280 

4o  57  5i.6* 

14.29 

i59 

963 

26  Persei,               /3 

2l 

58  25.56* 

3.871 

49  37  34.0* 

i4.3o 

160 

967 

27  Persei,               K 

5 

59  23.  96* 

4.oo9 

45  42  54.2* 

14.09 

161 

98i 

28  Persei,                w 

5 

3     i  37.41 

+  3.845 

5a  57  44.3 

i4-  10 

162 

982 

Hydri,                6 

5 

i   58.36 

—  0.008 

162  29  27.7 

i3.49 

i63 

986 

57  Arietis,               6 

4 

3     3.58* 

+3.4i8 

70  5o  39.9* 

i4*oo 

1  64 

997 

12  Eridani,              a 

31 

5  42.08* 

2.55o 

119  34  53.o* 

i4.46 

i65 

999 

58  Arietis,              £ 

5 

6  17.21* 

3.434 

69  3o  53.9* 

i3.73 

166 

IOOI 

Cassiopeae, 

5 

6  5i.i5* 

5.i52 

24  54    8.5* 

13.71 

167 

ioi3 

1  3  Eridani,              £ 

4 

8  33.o8* 

2.9IO 

92     22    48.8* 

i3.66 

1  68 

1028 

96  Ceti,                   *' 

5 

II    29.9I 

3.139 

87  ii     3.o 

13.43 

169 

io34 

6  1  Arietis,              r1 

5 

12    34.43* 

3.449 

69  23  5o.2* 

i3.33 

170 

1037 

1  6  Eridani,              r* 

31 

12    50.67* 

+2.665 

112   18  24.8* 

i3.4i 

171 

io38 

Mensae,    • 

5 

12    52.  60 

—2.334 

169  32  55.2 

13.36 

172 

io43 

33  Persei,                a 

Si 

i3  38.35* 

+4.24i 

4o  4o  39.3* 

i3.27 

i73 

io44 

Eridani, 

41 

i3  55.  o5 

2.365 

i33  38  48.6 

14.04 

1  74 

io56 

Hydri, 

5 

16  i8.9o 

0.673 

157  28  29.7 

12.66 

i75 

1057 

i  Tauri,                 o 

41 

16  44.83 

3.222 

81   3o  10.7* 

i3.o5 

176 

io58 

Camelopardi, 

4 

16  57.  92* 

4.789 

3o  35   16.6* 

i3.i3 

177 

1062 

Camelopardi, 

4 

17  58.42* 

4.725 

3i   38  49.0* 

i3.o6 

178 

io65 

Camelopardi, 

5 

18  35.65 

4.523 

35     4  25.o 

12.  87 

179 

1068 

2  Tauri,                 f 

4 

19     2.68* 

+3,243 

80  47  37.8* 

12.02 

180 

1070 

Hydri, 

5 

19  44.O2 

—  1-794 

167  55  57.7 

i3.i5 

181 

1071 

35  Persei,               o 

5 

20       I.3l* 

+4.194 

42  3i    38.7* 

I2.90 

182 

io9o 

17  Eridani, 

41 

23  10.75 

2.973 

95  35  36.6 

12.65 

i83 

io99 

37  Persei,                ^ 

5 

25    5l.O2* 

4.228 

42  18  4o.8* 

12.45 

i84 

IIOO 

1  8  Eridani,              e 

31 

25    52.25 

2.826 

99  58     8.3* 

12.43 

i85 

u  04 

I9  Eridani,              r5 

4 

27    9.79 

2.643 

112       8    20.3 

12.38 

186 

III2 

10  Tauri, 

41 

29  i3.43 

3.o58 

90     4  40.9 

n.74 

187 

1125 

Eridani, 

5 

3i  42.77 

2.147 

i3o  46  10.0 

12.08 

188 

1129 

39  Persei,                J 

3 

32  i5.85* 

4.235 

42  4  i   5o.o* 

12.02 

189 

n33 

Camelopardi,     « 

5 

32  58.  06* 

5.i6i 

27     8     6.4* 

ii.  93 

190 

u37 

Camelopardi,     y 

41 

34  34.  89 

6.189 

19     8  i4.5 

11.86 

191 

u38 

38  Persei,                o 

4 

34  55.48* 

3.733 

58  ii   27.3* 

11.89 

192 

n39 

4  1   Persei,                v 

4 

35     i.i3* 

4.o46 

47  54     i.5* 

11.86 

i93 

n44 

Camelopardi, 

5 

35  50.07 

5.392 

24  56  42.6 

11.79 

194 

1147 

17  Tauri, 

41 

35  58.74* 

3.548 

66  21  43.9* 

11.76 

i95 

u48 

e3  Eridani,              d 

31 

36    4-o5 

2.871 

100  i  6  28.0* 

12.  5l 

196 

u5o 

Eridani, 

5 

36  17.03 

2.378 

122    25     l3.5 

n.  73 

197 

n5i 

i9  Tauri, 

5 

36  17.22* 

3.557 

66     o  28.0* 

ii.  75 

198 

n54 

20  Tauri, 

5 

36  54.  5o* 

3.555 

66     6   19.0* 

11.68 

199 

n59 

Eridani,              vl 

5 

37  I7.'o5 

2.248 

127  47  19.6 

ii.  73 

200 

1161 

23  Tauri, 

5 

3  37  25.92* 

+  3.549 

66  3i   23.4* 

—  11.66 

TABLE   XXX. 


407 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c         |           d 

a' 

b' 

c' 

d' 

161 

+8.9376 

+8.9160 

+o.6477 

+8.8568 

+9.5i88 

—9.7792 

—  1.1622 

+9.8384 

i5a 

8.6845 

8.6634 

o.495i 

+7.4699 

—9.5923 

—8.6452 

.  1620 

9.8387 

i53 

8.6866 

8.6689 

o.4678 

—7-8449 

-9.7235 

+9.0164 

.1604 

9.84o5 

1  54 

8.7866 

8.7698 

o.58oo 

+8.5784 

+8.9469 

—9.6495 

.1699 

9.84io 

i55 

8.72i3 

8.7053 

o.4a38 

—8.3342 

—9.8328 

+9.4703 

.1696 

9.84i3 

i56 

9.0497 

9.o343 

o.o45o 

—  9-oo58 

—9.9103 

+9.8i32 

.1593 

9.8416 

i57 

8.6836 

8.6719 

o.4678 

-7.8374 

—9.7234 

+9.0090 

.i575 

9-8436 

168 

8.8600 

8.8534 

0.6182 

+8.738i 

+9.3899 

—9.7308 

.i55o 

9.8462 

i59 

8.7945 

8.7885 

o.5876 

+8.6o59 

+9.0945 

—9.6639 

.i54? 

9.8465 

160 

8.8197 

8.8174 

0.6010 

+8.6637 

+9.2694 

—  9.6946 

,x5a8 

9.8483 

161 

8.7800 

8.7862 

0.5846 

+8.5793 

+9.0469 

—  9.6456 

.i486 

9.8525 

162 

9.i9i3 

9.1988 

8.6822 

—9.1707 

—  9.9089 

+9.825i 

.i479 

9.8532 

i63 

8.6923 

8.7039 

o.53i9 

+8.2o83 

—  9.2425 

-9.3597 

.i458 

9-8552 

1  64 

8.7230 

8.  7446 

o.4oi5 

—  8.4i64 

—9.8664 

+9.53i8 

.1406 

9.8600 

i65 

8.6895 

8.7i34 

o.5357 

+8.2335 

—9.1827 

—  9.38i3 

.i395 

9.8611 

1  66 

9.0357 

9.0616 

0.7132 

+8.9933 

+9.6790 

-9.7938 

.i384 

9.8621 

167 

8.6624 

8.6949 

o.4637 

—  7.8746 

-9.7385 

+9.o448 

.i349 

9.865i 

168 

8.65io 

8.6946 

o.494i 

+  7.3423 

—  9.5988 

—  8.5i79 

.1288 

9.8702 

169 

8.6769 

8.7247 

0.5372 

+8.2233 

—  9.i587 

—  9.3707 

.1265 

9.8721 

170 

8.68x4 

8.7302 

+0.4261 

—8.2607 

—  9.8354 

+9-4o3o 

.1259 

9.8726 

171 

9.3889 

9.4378 

—o.368i 

-9.8817 

—  9.9o96 

+9.8164 

.1259 

9.8726 

172 

8.83i8 

8.8837 

+0.6268 

+8.7117 

+9.4538 

—9.7019 

.1243 

9.8739 

i73 

8.7859 

8.8387 

0.3255 

—8.6248 

—  9.9i95 

+9.6604 

.I237 

9.8743 

1  74 

9.0669 

9.  1189 

9.8026 

—  9.0224 

—  9.9366 

+9.7818 

.n85 

9.8783 

i75 

8.644o 

8.7077 

o.5o8i 

+  7.8i36 

—  9.6016 

—8.9849 

.  n75 

9.8790 

176 

8.9321 

8.9967 

0.6798 

+8.8671 

+9.6348 

—9.7498 

.1171 

9.8794 

177 

8.9166 

8.9860 

0.6739 

+8.8467 

+9.6228 

—9.7427 

.n48 

9.8810 

178 

8.8757 

8.9465 

0.6553 

+8.7887 

+9.5732 

—9.7242 

.u34 

9.8820 

179 

8.6398 

8.7123 

+o.5ioo 

+7.8438 

—  9.4862 

—9.0143 

.1124 

9.8827 

180 

9.3i23 

9.3875 

—0.2344 

—  9.3026 

—  9.9238 

+9.7990 

.  1109 

9.  8838 

181 

8.8020 

8.8783 

+0.6219 

+8.6695 

+9.4335 

—  9.6755 

.  IIO2 

9-8843 

182 

8.6268 

8.7i52 

0.4726 

—7.6157 

—  9.7o54 

+8.7897 

.  io3o 

9.8892 

188 

8.7903 

8.8891 

0.6253 

+8.6592 

+9.4568 

—  9.6634 

.096-7 

9.8933 

184 

8.625o 

8.  7239 

0.4604 

—  7.8634 

—  9.75o4 

+9.o328 

.0967 

9.8934 

i85 

8.6485 

8.7625 

0.4221 

-8.2247 

-9.8439 

+9.3675 

.o936 

9.8953 

186 

8.6io3 

8.7223 

o  4870 

-5.7446 

—9.6386 

+6.92o6 

.0886 

9.8983 

187 

8.7248 

8.8466 

0.3326 

—  8.5397 

—9.9293 

+9.595i 

.0824 

9.9019 

188 

8.77i4 

8.8953 

0.6262 

+8.6377 

+9.4672 

—  9.645i 

.0810 

9.9027 

189 

8.9419 

9.0686 

0.7127 

+8.8912 

+9.7119 

—  9.7264 

.O792 

9.9037 

190 

9.0812 

9.2i43 

0.7906 

+9«o565 

+9.7914 

—  9.7482 

.075l 

9.9060 

191 

8.6666 

8.8011 

0.6728 

+8.3885 

+8.7497 

—9.4939 

.0742 

9.9066 

192 

8.7253 

8.  8601 

0.6069 

+8.55:6 

+9.3399 

—9.5981 

.0740 

9.9066 

i93 

8.9685 

9.1066 

0.7320 

+8.9260 

+9-74o5 

—9.7271 

.0719 

9.9077 

194 

8.63i3 

8.7699 

0.5495 

+8.2343 

—8.8645 

—9.3724 

.o7i5 

9.9079 

196 

8.6000 

8.739o 

0.4586 

-7.85i3 

—  9.7569 

+9.0204 

.0713 

9.9081 

196 

8.6660 

8.8o59 

o.377i 

—  8.3953 

—9.9033 

+9.4978 

.0707 

9.9084 

197 

8.63i6 

8.77i5 

o.55o5 

+8.2408 

—8.8274 

-9.3777 

.0707 

9.9084 

198 

8.6297 

8.7-720 

o.55o4 

+  8.2372 

—8.8338 

—9.3744 

.0691 

9.9092 

199 

8.6920 

8.8359 

o.348o 

-8.4793 

—9.9247 

+9-5531 

.0681 

9-9°97 

200 

+8.6269 

+8.77i4 

+0.5493 

+  8.2272 

—8.8704 

—9.3658 

—   .0677 

+9'9°99 

408 


TABLE  XXX. 


CATALOGUE  OP  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.   m.        s. 

s. 

o         /•            // 

// 

2OI 

1166" 

25  Tauri,                v 

3 

3  38  34.56* 

+  3.552 

66  21  46.7* 

—  11.56 

202 

1168 

26  Eridani,              ?r 

5 

39     3.23 

2.83o 

102    34    32.2 

n.  61 

203 

1174 

3o  Tauri,                 e 

5 

4o     2.99 

3.279 

79  19  20.7* 

1  1  .45 

2O4 

1176 

27  Tauri, 

5 

4o  15.09* 

3.552 

66  24  35.o* 

H.44 

205 

n8i 

27  Eridani,             r8 

4* 

4o  23.78 

2.582 

n3  4i  43.o 

io.97 

206 

1191 

28  Eridani,              r7 

5 

4i   12.  59* 

2.577 

ii4  20  34.4* 

n.45 

207 

1197 

Reticuli, 

4 

42  19.71 

0.703 

i55  16  46.5 

11.47 

208 

1199 

Eridani, 

5 

43    3.90 

2.220 

I28     4  5i.8 

n  .4o 

209 

I2OI 

Eridani,             v* 

4 

43  5o.3o 

2.233 

126  39  26.5 

ii.  i5 

2IO 

1203 

Camelopardi, 

5 

44  i3.8i* 

5.197 

27    22    26.5* 

II  .25 

211 

I2O7 

44  Persei,               C 

II 

44  42.86* 

3.755 

58  33  59.4* 

n  .14 

212 

I2II 

Cassiopeae, 

5* 

45  i3.24* 

4-9.565 

9  43  36.5* 

ii.  18 

213 

I2l5 

Hydri, 

5 

46   10.10 

—0.445 

162  23  54.8 

ii  .06 

214 

1216 

32   Eridani, 

5 

46  45.82 

+3.  on 

93  24     6.8 

II  .01 

2l5 

1217 

33  Eridani,             r" 

5 

47  20.24 

2.556 

ii5     3  4o.4 

10.88 

216 

1219 

45  Persei,               e 

3* 

47  48.  o5* 

3.997 

5o  25  43.6* 

10.91 

217 

1220 

Eridani,             v* 

5 

47  55.89 

2.277 

125     IO    44-4 

10.88 

218 

1228 

46  Persei,               | 

5 

49  14.57 

+3.869 

54  38  43.9 

10.79 

219 

1230 

Hydri,                7 

3 

49  39.76 

—  i.o38 

i64  4i   53.o 

10.87 

220 

1234 

34  Eridani,             7 

it 

5i     1.93* 

+2.796 

io3  56  19.7* 

1  0.60 

221 

1241 

35  Tauri,                3, 

4 

52  22.49* 

3.3i5 

77  56  i5.8* 

10.  61 

222 

1243 

36  Eridani,             r9 

5 

53  3i.92 

2.555 

n4  26  42.0 

IO.52 

223 

1245 

35  Eridani, 

5 

53  56.19 

3.o34 

91   58  27.3 

10.43 

224 

I25l 

38  Tauri,                 v 

5 

55  10.90 

3.i85 

84  25  5i.o 

io.38 

225 

1264 

47  Persei,               A 

4* 

55  25.82* 

4.43i 

4o     3  4i.5* 

10.  3i 

226 

1257 

37  Tauri,                A1 

5 

55  5o.o3* 

3.534 

68   19*57.2* 

10.28 

227 

1259 

Reticuli,             <J 

5 

56  23.  4i 

0.954 

i5i  49  34.7 

9.99 

228 

1266 

48  Persei,                c 

5 

57  47-24* 

4.326 

42  4i    36.9* 

IO.I2J 

229 

1270 

Reticuli,             7 

5 

58  44.36 

o.83i 

i52  34  42.5 

10.24 

230 

1271 

Reticuli,             i 

5 

58  54.  o5 

0.986 

i5i   3o     3.7 

9.95 

23l 

1287 

5  1   Persei,               /a 

4i 

4     3  53.99* 

4.372 

4i   58  39.o* 

9.67 

232 

1290 

38  Eridani,              ol 

** 

4  32.77 

2.922 

97  i3  58.o 

9-74 

233 

1291 

52  Persei,               / 

5 

4  4i.4g 

4.o59 

49  54     8.8 

9.60 

234 

1299 

Horologii,          d 

5 

5  47.66 

2.027 

i32  23   i4.5 

9.65 

235 

i3oi 

Persei,               b1 

5 

6  58.98 

4.48i 

4o    4  47.0 

9.44 

236 

i3o3 

39  Eridani,             A 

5 

7  i5.73 

2.85i 

100  37  57.8 

9-29 

237 

i3o4 

49  Tauri,                // 

5 

7  23.  5i 

3.25o 

81  29  14.2 

9-44 

238 

1  309 

4o  Eridani,             o2 

41 

8  22.16 

2.763 

97  53  23.5 

5.  94 

239 

i3i5 

Horologii,          a 

5 

9       2.23 

1.992 

i32   39  57.2 

9.19 

24o 

i326 

52  Tauri,                 0 

5 

ii     8.i5 

3.673 

63     o  44-3 

9.i3 

241 

i328 

54  Tauri,                7 

H 

ii    i5.75* 

3.4o7 

74  44  21.  i* 

9.i4 

242 

i33i 

Doradus,            7 

4 

12     6.08 

1.556 

i4i  52     0.4 

9.34 

243 

i333 

4  1  Eridani,             v* 

3* 

12    12.96* 

2.264 

124  10     6.4* 

9-°7 

244 

i336 

Reticuli,             a 

3* 

12  3o.33 

0.745 

i52  5i     4.2 

9.o4 

245 

1  344 

Reticuli,             e 

5 

i3  54.12 

i  .008 

149  39  53.9 

8.58 

246 

1  346 

6  1   Tauri,                6l 

4 

i4  17.36* 

3.45o 

72  48  5o.i* 

8.91 

247 

1  348 

Horologii, 

5 

i4  32.37 

1.907 

i34  37  48.9 

8.82 

248 

i356 

64  Tauri,                 d8 

44 

i5  27.26* 

3.45i 

72  54  29.2* 

8.83 

249 

i358 

Reticuli,            0 

5 

16     0.90 

o.683 

i53  37  10.4 

9.o3 

260 

1  365 

68  Tauri,                 d3 

5 

4  16  49.01* 

+3.464 

72  25     9.1 

-  8.74 

TABLE   XXX. 


409 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b        \         c 

d 

a' 

b'                  c' 

d' 

201 

+8.6244 

+8.7734 

+o.55oo 

+8.2275 

—8.8488 

—9.  3655 

—    .0646 

+9.9115 

202 

8.5956 

8.7466 

o.45i3 

—7.9335 

—9.7791 

+9.0991 

.o634 

9.9121 

203 

8.5900 

8.7449 

o.5i55 

+7.8578 

-9.4387 

—  9.0263 

.0607 

9.9134 

204 

8.6197 

8.7755 

o.55oi 

+  8.2220 

—8.8426 

—  9.3602 

.0601 

9.9i37 

205 

8.6197 

8.7761 

0.4l32 

—8.2238 

—9.86i5 

+9.36i6 

.o597 

9.9139 

206 

8.6197 

8.7793 

o.4io5 

—8.2848 

—9.8659 

+9-37o4 

.o575 

9.9150 

207 

8.9548 

9.  1190 

9.8292 

—  8.9130 

—  9.9689 

+9.7io5 

.o544 

9.9164 

208 

8.6780 

8.8452 

0.3433 

—8.4682 

—  9.9302 

+9.54o3 

.o524 

9.9174 

209 

8.6676 

8.838o 

o.35i4 

—8.4436 

—9.9257 

+9.524o 

.o5o2 

9.9184 

210 

8.9083 

9.0802 

o.7i7i 

+8.8567 

+9.73oi 

—9.6954 

.0492 

9.9189 

2H 

8.6384 

8.8i23 

o.574o 

+8.3557 

+8.7993 

—9.4628 

.o478 

9.9195 

212 

9  .  34o4 

9.5i63 

+0.9805 

+9.334i 

+9.8732 

—9.7379 

.o464 

9.9201 

til 

9.0848 

9.2647 

—9.6488 

—  9.0639 

—9.9668 

+9.72o6 

.o436 

9.9213 

2l4 

8.5644 

8.7467 

+0.4777 

-7.3377 

—  9.6838 

+8.5i3o 

.0419 

9.9221 

2l5 

8.6049 

8.7896 

o.4o6i 

—  8.23i8 

-9.8737 

+9-365o 

.o4o3 

9.9228 

216 

8.6736 

8.86o3 

0.6017 

+8.4778 

+9.2999 

—9.5409 

.o389 

9.9234 

217 

8.6478 

8.835o 

o.358o 

—  8.4o83 

—  9.9231 

+9.4968 

.o385 

9.9235 

218 

8.6449 

8.8376 

+0.5876 

+8.4o73 

+9.1242 

—  9.4949 

.o347 

9.9251 

219 

9.i338 

9.3280 

—  0.0197 

—  9.  1181 

—9.9671 

+9.7i56 

.o335 

9  .9256 

22O 

8.564o 

8.7642 

+0.4456 

—  7.9458 

-9.7958 

+9.  1089 

.0294 

9.9273 

221 

8.5567 

8.7625 

o.52o3 

+  7.8768 

—9.3927 

—  9.0432 

.0253 

9.9289 

222 

8.5842 

8.7949 

0.4071 

—  8.2010 

—9.8736 

+9.3363 

.0218 

9.9302 

223 

8.5425 

8.7549 

0.4816 

—7.0796 

—  9.6657 

+8.2555 

.02O5 

9.9307 

224 

8.54o4 

8.7581 

0.5027 

+7.5274 

—  9.5427 

—8.7014 

.0166 

9-9321 

225 

8.7289 

8.9477 

0.6465 

+8.6128 

+9.5748 

—9.5975 

.oi59 

9.9324 

226 

8.568i 

8.7886 

0.5472 

+8.I354 

—8.9390 

—9.2797 

.oi46 

9.9329 

227 

8.86o4 

9.o833 

9.9679 

-8.8o57 

—  9.9829 

+9.6558 

.0128 

9.9335 

228 

8.6988 

8.9277 

0.6353 

+8.565i 

+9.5298 

—9.5724 

.0084 

9.935i 

229 

8.8637 

9.0968 

9.9274 

—8.8120 

—9.9851 

+9.65i3 

.oo53 

9.9361 

230 

8.8478 

9.0816 

9.9749 

—8.7917 

—  9.9850 

+9.6465 

.oo48 

9.9363 

23l 

8.6845 

8.94o3 

o.64o5 

+8.5557 

+9-5561 

—9.5571 

0.9881 

9.9417 

232 

8.5m 

8.7698 

o.4657 

—  7.6111 

—9.7330 

+8.7837 

0.9859 

9.9424 

233 

8.6235 

8.8828 

0.6082 

+8.4324 

+9.366i 

—9.4922 

0.9854 

9.9425 

234 

8.6349 

8.8992 

o.3oo8 

—8.4636 

—  9.9603 

+9.5o8i 

0.9816 

9.9437 

235 

8.6903 

8  .9600 

o.65o3 

+8.574i 

+9.5962 

—9.559o 

0.9774 

9.9449 

236 

8.5o57 

8.7766 

0-.4548 

—7-77I7 

—9.7702 

+8.9403 

0.9765 

9.9452 

23? 

8.5o25 

8.774o 

o.5n5 

+7.6728 

—9.4752 

—8.8441 

o.976o 

9.9453 

238 

8.4988 

8.7743 

0.4634 

-7.6359 

—  9-74i3 

+8.8079 

o.9725 

9-9463 

23g 

8.6253 

8.9044 

0.2967 

—8.4564 

—9.9633 

+9.4989 

o.97oi 

9-947o 

240 

8.5342 

8.823o 

0.5654 

+8.1911  . 

+  8.2279 

-9.3i7i 

0.9625 

9.9490 

24l 

8.4993 

8.7887 

0.5309 

+  7.9196 

—  9.2662 

—  9.0801 

0.9620 

9.9492 

242 

8.6906 

8.9833 

o.  1911 

—  8.5857 

—9.9863 

+9.5524 

0.9589 

9.95oo 

243 

8.5624 

8.8563 

0.3544 

—  8.3119 

—9.9344 

+9.4o57 

0.9584 

9.9501 

244 

8.8198 

9.  n5o 

9.8721 

—8.7691 

—9.9964 

+9.6045 

0.9574 

9.9504 

245 

8.77o5 

9.0723 

o.onS 

—8.7065 

—9.9963 

+9.586o 

0.9521 

9.95i7 

246 

8.4922 

8.7958 

0.5367 

+  7.9627 

—  9.  1761 

—9.1190 

0.9507 

9.952I 

247 

8.6191 

8.9239 

0.2760 

—8.4658 

—9.9723 

+9.4941 

0.9497 

9.9523 

248 

8.4875 

8.7967 

0.5365 

+7.9557 

—9.1787 

—  9.  1122 

0.9462 

9.9532 

24g 

8.  8180 

9.  1299 

9.8114 

—8.7702 

—9.9991 

+  9.5940 

0.9440 

9.9537 

250 

+8.4833 

+8.7991 

+o.538i 

+7-9634 

—  9.i5i4 

—  9.1187 

—0.9409 

+9.9544 

410 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.       s. 

s. 

o        /         // 

// 

25l 

i367 

69  Tauri,                 vl 

5 

4   17   20.  3o* 

+3.579 

67  3i   53.i* 

—  8.66 

252 

1370 

?3  Tauri,                 TT 

5 

18     8.  ii 

3.38o 

75  37  48.5 

8.59 

253 

1372 

43  Eridani,             v* 

4 

18  24.35 

2.254 

124    22     1  0.0* 

8.59 

254 

i376 

74  Tauri,                 e 

H 

19  51.70* 

3.494 

71     9  25.i* 

8.47 

255 

i38o 

11  Tauri,                 0l 

41 

20     o.55* 

3.4i3 

74  22  3i.5* 

8.47 

256 

i38i 

78  Tauri,                 62 

4* 

20       6.18* 

3.420 

74  28     o.o* 

8.46 

257 

i383 

Reticuli,             J? 

5 

20    16.91 

o.6i9 

i53  44  34-4 

8.79 

258 

1409 

86  Tauri,                p 

5 

25     20.  4l* 

3.402 

75  28  33.o* 

8.02 

259 

i4i3 

Cedi,                   6 

5 

26     l4-77 

i.84i 

i35   16  41.7 

8.08 

260 

1419 

47  Eridani, 

5 

26  58.23* 

2.888 

98  32  57.7* 

7.92 

261 

l42O 

87  Tauri,                a 

i 

27   19.12* 

3.436 

73  47  49.0* 

7-74 

262 

1421 

88  Tauri,                 d 

5 

27  24.85 

3.285 

80     9     4.7 

7.83 

263 

1422 

5o  Eridani,             t'6 

4i 

27  37.56 

2.35i 

I2O       4    20.5 

7-64 

264 

1429 

48  Eridani,             v 

4 

28  49.69 

2.994 

93  39  46.8 

I'll 

265 

i433 

52  Eridani,             v"* 

3^ 

29  43.4o 

2.333 

I2O    52    23.7* 

7.67 

266 

1434 

90  Tauri,                 c1 

5 

29  46.75 

3.35o 

77  47  39.8 

7.68 

267 

i438 

Doradus,             a 

3 

3o  45.8i 

1.289 

i45  21  25.4 

7.61 

268 

i44i 

53  Eridani, 

4 

3i    18.87 

2.746 

io4  36     3.o 

7.41 

269 

1442 

93  Tauri,                 c2 

5 

3i  42.69* 

3.336 

78     6     1.2 

7.57 

270 

1449 

94  Tauri,                 r 

5 

33   i4.84* 

3.592 

67  20     8.4* 

271 

i45i 

54  Eridani, 

4 

33  53.01 

2.623 

109  57  48.3 

7.25 

272 

i456 

4  Camelopardi, 

5 

35  3i.65* 

4.96o 

33  3o  58.o* 

7.09 

273 

i458 

Caeli,                   a 

4* 

35  43.95 

i.93o 

1  32       9       9.0 

7.16 

274 

1  464 

Cseli,                   /? 

5 

36  45.55 

2.  122 

127  26  25.3 

7.32 

2?5 

1469 

57  Eridani,             fj. 

5 

38     0.43 

S.ooi 

93  32     o.3 

7.o3 

276 

1473 

Pictoris,             A 

5 

38  55.97 

i.sii 

i4o  45  55.i 

7.06 

277 

1474 

9  Camelopardi,     a 

4 

39     9.42* 

5.9o6 

23  55   i3.7* 

6.95 

278 

i486 

i  Orionis,              TTI 

4 

4i  42.  i5 

3.258 

83  18  18.7 

6.71 

279 

i49i 

2  Orionis,              ?r2 

5 

42  26.61 

3.272 

81   21  4i  -7 

6.63 

280 

i4g5 

3  Orionis,              TT* 

4 

43  i3.3o 

3.i94 

84  39  22.8 

6.56 

281 

i5oo 

4  Orionis,              o1 

5 

44    3.o6* 

3.388 

76     o  i4.8* 

6.47 

282 

i5o4 

7  Camelopardi, 

5 

45  16.57 

4.785 

36  29  4i  .4 

6.42 

283 

i5o7 

6  1   Eridani,             w 

5 

45  31.69 

2.946 

95  42  28.7 

6.4i 

284 

i5i4 

8  Orionis,               ?r5 

4i 

46  26.  5i 

3.122 

87  48  32.4 

6.33 

285 

l52O 

3  Aurigae,              L 

4 

47   i3.87* 

3.895 

57     4  37.7* 

6.25 

286 

i525 

9  Orionis,              o2 

5 

47  56.48* 

3.37o 

76  43  38.o* 

6.16 

287 

i53o 

4  Aurigae, 

5 

49    4.77 

4-4.o58 

52  20  3o.8 

6.02 

288 

i532 

Mensae, 

5 

49  25.09 

—  2.322 

i  66  34  29.9 

5.89 

289 

i536 

10  Camelopardi,     /3 

4£ 

5o     5.72 

+5.3o3 

29  47     5.2* 

6.00 

290 

i54o 

7  Aurigae,              e 

4 

5i   12.  9i* 

4.29I 

46  24  16.7* 

5.93 

291 

i54i 

8  Aurigae,              f 

4 

52       O.II* 

4.181 

49     8  56.  9* 

5.86 

292 

1  544 

63  Eridani, 

5 

52  44.67 

2.840 

100  29   12.7 

5.68 

293 

1  546 

ii  Camelopardi, 

5 

53     7.18 

5.i84 

3i    i4  39.5 

5.76 

294 

i55i 

102  Tauri,                 L 

4* 

54     8.02* 

3.58i 

68  37  45.8* 

5.65 

295 

i552 

65  Eridani,             ^ 

5 

54   10  09 

2.9o6 

97  23  52.8 

5.69 

296 

i554 

9  Aurigae, 

5 

54  56.42 

4.678 

38  36  27.7 

5.47 

297 

i557 

1  1  Orionis, 

5 

56     0.07* 

3.424 

74  48  34-  9* 

5.49 

298 

i558 

10  Aurigae,              ij 

4 

56     o.3o* 

4.195 

48  58  28.2* 

5.48 

299 

i559 

Leporis, 

5 

56     3.90 

2.438 

116  29  26.  7 

5.44 

3oo 

i565 

Camelopardi, 

5 

4  57  55.  o3 

4-9-671 

10  57  24.2 

—  5.42 

TABLE   XXX. 


411 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of                                 ||                                  Logarithms  of 

a 

b                c                  d 

a' 

b' 

c' 

d' 

25l 

+8.4948 

+8.8i3i 

+0.5526 

+8.0771 

—  8.752i 

—  9.2189 

—  o.9388 

+9.9549 

262 

8.4712 

8.7933 

0.5289 

+  7.8659 

—  9.2929 

—  9.0282 

o.9357 

9.9556 

253 

8.5396 

8.863i 

o.35n 

—  8.29i3 

—  9.9386 

+9.384i 

o.9346 

9.9559 

254 

8.4743 

8.8o5o 

o.542i 

+7.9835 

—  9.0723 

—  9.i357 

0.9287 

9.9572 

255 

8.4662 

8.7976 

0.5327 

+  7.8965 

—  9.2401 

—  9.o562 

o.928i 

9.9573 

256 

8.4656 

8.7974 

0.5324 

+7.8934 

—  9.2438 

—  9.o534 

0.9.278 

9.9574 

257 

8.8029 

9.i356 

9'7873 

—  8.7556 

—  O.O223 

+9-5775 

0.9270 

9.9575 

258 

8.44i6 

8-7999 

o.  5299 

+  7.84o9 

-9-2797 

—  9.0020, 

0.9058 

9.96i9 

25g 

8.5762 

8.939i 

0.2629 

-8.4277 

—  9.9811 

+9.45i2 

0.9018 

9  .9626 

260 

8.  4a52 

8.  -7920 

o.46o3 

—  7.5974 

—  9.7526 

+8.7686 

0.8987 

9.9632 

261 

8.4364 

8.8o5o 

o.535o 

+  7.8821 

—  9.2047 

—  9.o4o6 

o.897i 

9.9635 

262 

8.4248 

8.7939 

o.5i64 

+7.6580 

—9.4319 

—8.8276 

0.8967 

9.9636 

263 

8.48o3 

8.85o4 

0.3726 

—8.1802 

—  9.9224 

+9.2935 

o.8958 

9.9637 

264 

8.4i3o 

8.7895 

0.47^9 

—7.2184 

—  9.6920 

+8.3936 

o.89o4 

9.9647 

265 

8.4744 

8.8557 

0.3678 

—8.1846 

—  9.9275 

+9.2943 

0.8863 

9.9654 

266 

8.4i77 

8.7993 

0.5235 

+7.7428 

—  9.3597 

—  8.9o9o 

0.8861 

9.9654 

267 

8.6485 

9.o354 

o.  1074 

—8.5638 

—  o.oo34 

+9-4946 

0.8816 

9.9662 

268 

8.4i5o 

8.8o48 

0.4390 

—  7.8i65 

—  9.8i5i 

+8.9783 

o.879o 

9.9669 

269 

8.4o83 

8.8oo3 

0.5227 

+  7.7226 

—  9.3683 

—  8.8892 

0.8772 

9.9669 

270 

8.4265 

8.8269 

o.555o 

+8.0123 

—8.6274 

—  9.i535 

o.8699 

9.968i 

271 

8.4i55 

8.8i94 

0.4181 

-7.9488 

—9.8606 

+9-°979 

o.8669 

9.9686 

272 

8.6385 

9.o5i6 

0.6949 

+8.5595 

+9.73o9 

—9-4777 

o.8589 

9.9698 

2?3 

8.5o95 

8.9238 

0.2881 

—8.3363 

-9.9768 

+9.3824 

o.8579 

9.97oo 

274 

8.4747 

8.8948 

o.325o 

—8.2585 

—  9.9604 

+9.3344 

0.8528 

9-97°7 

275 

8.369o 

8.7963 

0.4762 

—7.1688 

—  9.69o9 

+8.334i 

0.8465 

9.97i6 

276 

8.5624 

8.995i 

o.  1860 

—  8.45i5 

—  o.ooo5 

+9.4286 

o.84i8 

9.9722 

277 

8.7543 

9.i884 

0.7708 

+8.7i53 

+9.8337 

—  9.4993 

o.84o5 

9-9724 

278 

8.3520 

8.  8010 

o.5o77 

+7.4186 

—  9.5o68 

—  8.59i7 

0.8273 

9.9742 

279 

8.35oo 

8.8o35 

o.5i35 

+  7.5266 

—  9.4583 

—  8.6978 

0.8233 

9.9746 

280 

8.3427 

8.  8010 

o.5o36 

+7.3u8 

-9.5367 

—8.4860 

0.8191 

9.9752 

281 

8.3494 

8.8i27 

0.5297 

+7-7320 

—  9.2842 

—  8.8959 

o.8i46 

9-9757 

282 

8.555i 

9.O26l 

0.6797 

+8.46o3 

+  9.7020 

—  9.4io8 

0.8078 

9.9765 

283 

8.33o2 

8.8o27 

o.4689 

—7.3279 

—  9.72II 

+8.5oi8 

o.8o64 

9-9767 

284 

8.3232 

8.8oi5 

o.494o 

+6.9o56 

—  9.599i 

—  8.o8i4 

0.8012 

9-9772 

285 

8.3944 

8.8777 

o.59o3 

+8.1296 

+9.i8i8 

—  9.2296 

o.7967 

9-9777 

286 

8.326o 

8.8i38 

0.5277 

+7.6870 

—  9.3o98 

—  8.85i3 

0.7925 

9.9782 

287 

8.4090 

8.9o42 

+0.6078 

+8.i95o 

4-9-377I 

—  9.2696 

o.7859 

9-9789 

288 

8.9397 

9.4372 

—0.3555 

—  8.9277 

—  o.on9 

+9-4696 

o.7839 

9-9791 

289 

8.6o53 

9.  io73 

+0.7242 

+8.5438 

+9.7869 

—  9.4160 

o.7798 

9.9795 

290 

8.4348 

8.9442 

0.6321 

+8.2734 

+9.5371 

—  9.3o93 

0.7730 

9.98oi 

291 

8.4,111 

8.9257 

0.6208 

+8.2267 

+0.4731 

—9.2816 

o.  7681 

9.9806 

292 

8.2925 

8.8i23 

0.4524 

—7.5526 

-9.7788 

+8.7214 

o.7635 

9.9810 

293 

8.5679 

9.o9o3 

0.7144 

+8.4999 

+9-7736 

—  9-39°9 

0.76l2 

9.98l2 

294 

8.3073 

8.8367 

o.553o 

+7.8689 

-8.7372 

—9.0141 

o.7547 

9.98i8 

295 

8.2798 

8.8094 

o.463o 

—7.3896 

—  9.7434 

+8.562I 

0.7545 

9.98i8 

296 

8.4760 

9.0110 

0.6701 

+8.3689 

+9.68o7 

—9.3402 

o.7495 

9.9823 

297 

8.2797 

8.8222 

o.534o 

+7.6980 

—  9.2225 

—8.8587 

0.7426 

9.9828 

298 

8.3866 

8.9292 

0.6221 

+8.2o38 

+9.4823 

—  9-2575 

0.7425 

9.9828 

299 

8.3i20 

8.855o 

0.3856 

—7.9614 

-9.9I23 

+9.o893 

0.7421 

9.9829 

3oo 

+8.9725 

+9.5289 

+o.9879 

+8.9645 

+9.9363 

-9.4195 

—0.7297 

+9.9839 

412 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

,  Annual 
Variation. 

h.   m.         s. 

s. 

3oi 

i573 

Caeli,                  7l 

5 

4  Sg    0.80 

+2.i53 

126  4i  29.3 

—    5.20 

302 

i575 

2  Leporis,             e 

4 

59     6.  74* 

-j-2.538 

112    34    34.1* 

5.22 

3o3 

z587 

Mensae, 

4* 

5     o     o.i5 

—  1.533 

i65   10     3.i 

4.91 

3o4 

i588 

67  Eridani,             /? 

3 

o  28.73 

+2.  948 

95  17     4.7 

5.07 

3o5 

1691 

1  5  Orionis, 

5 

I     7.o4* 

3.43o 

74  35  56.4* 

5.12 

3o6 

1697 

69  Eridani,             7, 

4 

i  58.26 

2.872 

98  57     1.7 

5.oo 

Soy 

1600 

Doradus,             £ 

5 

2  56.53 

i  .001 

147  4o  45.2 

5.io 

3o8 

1602 

1  1   Aurigae,              /j. 

5 

3  10.09 

4.094 

61  4i  55.8 

4.86 

Sog 

1608 

3  Leporis,             t 

4^ 

5  18.17 

2.798 

102        3     I2.7 

4.72 

3io 

1611 

i7  Orionis,              p 

5 

5  27.16 

3.i35 

87   19   18.7 

4.72 

3n 

1612 

Doradus,            fi 

5 

5  35.19 

o.5i4 

162     o     9.7 

4.04 

3l2 

i6i3 

1  3  Aurigae,               a 

i 

5  36.97* 

4.422 

44    9  39.4* 

4.3o 

3i3 

1614 

1  4a  Aurigae, 

5 

5  38.  73* 

3.899 

67  29  3o.o* 

4.73 

3i4 

1616 

5  Leporis,             p 

5 

6  ii.  59* 

2.688 

106  23   12.3* 

4.66 

3i5 

1617 

4  Leporis,             K 

5 

6  i8.39 

2.770 

io3     7  21  .3 

4.64 

3i6 

1623 

19  Orionis,              /? 

i 

7  19.86* 

2.884 

98    22    45.2* 

4.56 

3i7 

i63i 

1  5  Aurigae,              A 

5 

8  35.  61* 

4.210 

5o     2  26.4* 

3.8o 

3i8 

i638 

20  Orionis,              r 

4 

10  19.58 

2.916 

97     o  39.i 

4.28 

3i9 

i65o 

Columbae,           o 

5 

12     4.55 

2.  162 

125        2    46.1 

3.66 

320 

i653 

6  Leporis,              A 

4^ 

12    39.98* 

+  2.764 

103     20        8.2 

4.09 

321 

i659 

Doradus,            0 

5 

i3  53.  59 

—  O.075 

167  21    i5.8 

4.07 

322 

i665 

23  Orionis,              m 

5 

i4  57.i2* 

+  3.I52 

86  36   i6.3* 

3.9i 

323 

1672 

Pictoris,             C 

5 

i5  4i.4a 

i.457 

i4o  46     8.3 

4.  n 

324 

1681 

us  Tauri,                 ,3 

2 

16  48.  80* 

3.79i 

61   3i   29.5* 

3.57 

325 

1  684 

28  Orionis,              TJ 

4* 

16  56.22 

3.oi4 

92     32     22.6 

3.73 

326 

1687 

24  Orionis,              y 

2 

17     5.3i* 

3.220 

83  47  27.7* 

3.72 

327 

1690 

24  Aurigae,              <£ 

5 

17  4s.  5i 

3.974 

55  39  27.9 

3.63 

328 

i695 

n4  Tauri,                 o 

5 

18  37.61* 

3.602 

68   ii  47.5* 

3.63 

320 

1700 

3o  Orionis,              T/>3 

5 

i  8  58.  87 

3.143 

87     2   18.8 

3.56 

33o 

1704 

Pictoris  ,             K 

5 

19  36.52 

I  .  I  12 

i46  16  60.9 

3.12 

33i 

I7o6 

Camelopardi, 

5 

19  45.79 

8.002 

i5    4    2.4* 

3.5o 

332 

i7i5 

9  Leporis,             /3 

4 

21  49.i9* 

2.572 

no  52  56.3 

3.25 

333 

i7i7 

3  1   Orionis, 

5 

22       7.09 

3.o46 

91    12  53.8 

3.26 

334 

I722 

32  Orionis,              A 

5 

22  45.58* 

3.208 

84  10   16.9* 

3.22 

335 

I723 

25  Aurigae,              x 

5 

22    57.71* 

3.905 

57  55  27.2* 

3.27 

336 

I73o 

34  Orionis,               6 

2 

24    20.72* 

3.066 

90  24  53.i* 

3.o7 

337 

i73i 

36  Orionis,              v 

5 

24  40.70 

2.905 

97  24  58.8 

3.o7 

338 

i739 

Columbae,           e 

4 

25  53.  3i 

2.l3l 

126  35     2.6* 

2.85 

339 

i74i 

1  1   Leporis,              a 

H 

26     6.96* 

2.648 

107  56     0.7* 

2.96 

34o 

i?48 

37  Orionis,              f 

4* 

26  35.26 

3.291 

80  37     0.9 

2.89 

34i 

1749 

39  Orionis,              A 

4 

26  52.73 

3.3o2 

80  10  18.4 

2.85 

342 

1756 

Columbae, 

5 

27  47.53 

2.013 

128  37  2i.5 

2.81 

343 

1759 

42  Orionis,              c 

5 

27  59.34 

2.958 

94  56  3o.2 

2.78 

344 

1762 

44  Orionis,              i 

3£ 

28     5.9i 

2.936 

96     o  45.2 

2.77 

345 

1765 

46  Orionis,               e 

2^ 

28    36.22* 

3.o44 

91    18     8.2* 

2.73 

346 

1766 

4o  Orionis,              <j>* 

4J- 

28  4o.i4* 

3.293 

80  47  44.6* 

2.42 

347 

i767 

123  Tauri,                 C 

3* 

28  40.97* 

3.585 

68  57  14.9* 

2.71 

348 

i768 

26  Aurigae, 

5 

29     o.35* 

3.846 

59  36     8.1* 

2.69 

349 

i78o 

48  Orionis,              a 

4 

3i   i3.o8 

3.010 

92  4i  28.2 

2.60 

35o 

i785 

49  Orionis,              d 

5 

5  3i   37.71* 

+  2  .902 

97  18     4.i* 

—  2.42 

TABLE  XXX. 


413 


CATALOGUE  OP  1500  STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

/;                    c. 

d 

a'        |         b' 

c' 

d' 

3oi 

+8.3342 

+  8.8987 

+o.33i2 

—  8.  i  002 

—9.9616 

+9.i859 

—  0.7221 

+9.9844 

302 

8.2778 

8.843o 

+o.4o38 

—7.8620 

—9.8871 

+9.oo34 

0.7214 

9.9845 

3o3 

8.8287 

9.4006 

—  0.2567 

—  8.8i39 

—  0.0190 

+9.3983 

0.7152 

9.9849 

3o4 

8.2354 

8.8110 

+0.4700 

—7.1996 

-9.7169 

+8.3739 

0.7118 

9.9852 

3o5 

8.2448 

8.8253 

0.5349 

+7.6690 

—  9.2082 

—8.8292 

0.7072 

9.9855 

3o6 

8.2280 

8.8i52 

o.4574 

—  7.4200 

—9.7629 

+8.59o8 

0.7010 

9.9859 

3oy 

8.4875 

9.0822 

0.0097 

—  8.4i44 

—  0.0216 

+9-3x85 

0.6939 

9.9864 

3o8 

8.3191 

8.9i57 

0.6122 

+8.1114 

+9.4i53 

—  9.  1822 

0.6922 

9.9865 

3oo 

8.2072 

8.8211 

o.446i 

—  7.5270 

—9.7975 

+8.6934 

0.6759 

9.9875 

3io 

8.  1969 

8.8120 

0.4957 

+6.8664 

—9.5889 

—  8.0420 

0.6747 

9.9876 

3n 

8.5a38 

9.1400 

9.7968 

—8.4697 

—  0.0261 

+9-3i74 

0.6737 

9.9876 

3l2 

8.3521 

8.9685 

o.6443 

+8.2079 

+9.5966 

—  9.2270 

0.6734 

9.9877 

3i3 

8.2690 

8.8856 

0.5909 

+7.9993 

+9.1942 

—9.1014 

0.6733 

9.9877 

3i4 

8.2086 

8.8298 

0.4295 

—  7.6590 

—  9.8393 

+8.8171 

0.6689 

9.9879 

3i5 

8.2012 

8.8234 

0.4421 

-7.5573 

—9.8o85 

+8.7219 

0.6680 

9.9880 

3i6 

8.1861 

8.8170 

0.4592 

—7.3496 

—9.7568 

+8.52H 

0.6597 

9.9884 

317 

8.2866 

8.9284 

0.6194 

+8.o943 

+9.4678 

—9.1549 

0.6494 

9.9890 

3i8 

8.  i  597 

8.8169 

0.4639 

-7.2463 

—9.7403 

+8.4191 

0.6348 

9.9897 

3i9 

8.2280 

8.9012 

o.333i 

—7.9871 

—  9.9624 

+9.o763 

0.6194 

9.9904 

320 

8.1477 

8.8265 

+o.44io 

—  7.5io6 

—9.8116 

+8.6749 

o.6i4i 

9.9907 

321 

8.539i 

9.2296 

—8.8420 

—  8.5o42 

—  o.oSoo 

+9.2658 

0.6029 

9.9912 

322 

8.n54 

8.8162 

+0.4981 

+6.8879 

—9.5742 

—8.o632 

0.5929 

9.9916 

323 

8.3o65 

9.0147 

o.i654 

—8.1956 

—  o.oi48 

+9.1727 

0.5858 

9.9918 

324 

8.i525 

8.8722 

o.5778 

+7.83o8 

+8.9385 

—  8.9509 

0.5748 

9.9922 

325 

8.0957 

8.  8166 

0.4789 

—  6.7422 

—9.6785 

+  7.9x79 

0.5736 

9.9923 

326 

8.0963 

8.  8188 

0.5070 

+  7.i3o4 

—  9.5120 

—  8.3o39 

0.5721 

9.9923 

327 

8.1707 

8.8997 

0.5986 

+  7.9221 

+9.2929 

—  9.ox5o 

o.5658 

9.9926 

328 

8.no3 

8.8490 

o.5559 

+  7.6802 

—8.5740 

—8.8240 

0.5564 

9.9929 

329 

8.o75o 

8.8175 

0.4968 

+  6.7882 

—9.5827 

-7.9637 

0.5527 

9.9930 

33o 

8.3234 

9.0727 

o.o4o8 

—8.2434 

—  0.0252 

+9.i639 

o.546i 

9.9932 

33r 

8.65i2 

9.4023 

0.9010 

+8.636o 

+  9.9232 

—  9.227O 

o.5444 

9.9933 

332 

8.0731 

8.8474 

0.4095 

—7.6261 

—  9.8791 

+  8.7716 

0.5219 

9.9940 

333 

8.o4o3 

8.8181 

0.4832 

—  6.3667 

—  9.6578 

+7.5427 

o.5i85 

9.9940 

334 

8.o35i 

8.8204 

o.SoSg 

+7.o4i8 

—9.52o5 

—  8.2i57 

o.5ux 

9.9942 

335 

8.1024 

8.8902 

0.5908 

+7.8275 

+9.1942 

—  8.93i6 

o.5o87 

9.9943 

336 

8.oi4i 

8.  8186 

o.485g 

—5.8740 

—9.6446 

+7.o5oi 

0.4924 

9.9947 

337 

8.0137 

8.8224 

0.4622 

—  7.1246 

-9.  7465 

+8.297o 

o.4884 

9.9948 

338 

8.o848 

8.9089 

0.3273 

—7.8496 

—9.9677 

+8.936o 

o.4733 

9.9952 

339 

8.oi38 

8.84o8 

0.4220 

—  7.5022 

—9.  8558 

+8.6566 

0.4704 

9.9952 

34o 

7.9919 

8.825! 

0.5171 

+7.2042 

—9.4264 

-8.3745 

o.4644 

9.9954 

34  1 

7.9887 

8.8258 

o.5i85 

+  7.2210 

—9.4128 

—  8.3906 

0.4606 

9.9955 

342 

8.o775 

8.9268 

o.3o38 

-7.8728 

—  9.9809 

+8.94i7 

o.4486 

9-9957 

343 

7.9692 

8.82i3 

0.4707 

—  6.9045 

—9.7140 

+8.0789 

0.4459 

9.9958 

344 

7.9685 

8.8221 

0.4670 

—6.9887 

—  9.7286 

+8.1624 

o.4444 

9.9958 

345 

7.9594 

8.8199 

o.483o 

—  6.3i59 

—9.6594 

+7.4919 

0.4376 

9.9909 

346 

7.964o 

8.8255 

o.5i66 

+7.1680 

—  9.43i2 

—8.3385 

o.4367 

9.9959 

347 

7.9881 

8.8498 

o.5539 

+7-5434 

—8.6920 

—  8.6895 

0.4365 

9-9959 

348 

8.0180 

8.884i 

0.5852 

+7.7221 

+9.1042 

—8.834o 

o.432o 

9.996o 

349 

7.9222 

8.8210 

0.4783 

—  6.5938 

—9.68x3 

+7.7695 

o.4ooo 

9.9966 

35o 

+  7.9190 

+8.8241 

+o.4625 

—  7.0231 

—  9.7455 

+8.x957 

—0.3938 

+9-9967 

414 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.  C 

Constellation. 

Mag. 

Right  Ascension. 
Jan.  1,  1850. 

Annual 
Variation 

North  Polar  Dist. 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.         s. 

s. 

35i 

1791 

Doradus,            ft 

4 

5    32    I9.75 

•4-0.  5o6 

l52    35    20.0 

—  2.45 

352 

i794 

5o  Orionis,              C 

2 

33  ii.  5i* 

3.o3o 

92      I    37.2* 

2.33 

353 

1802 

Columbae,           a 

2 

34  i3.i5* 

2.177 

124     9  28.4* 

2.25 

354 

1823 

1  3  Leporis,              y 

4 

38   12.  61* 

2.5oo 

112  3o     2.8* 

i.55 

355 

i83o 

29  Aurigae,              r 

5 

38  46.9-7 

4.i53 

5o  62  35.8 

i.79 

356 

i837 

1  32   Tauri, 

5 

39  48.  77* 

3.683 

65  29  17.9* 

i.75 

357 

i84o 

1  4  Leporis,              C 

41 

/  * 
40     9.70 

2.720 

io4  52  55.7 

I.72 

358 

i843 

53  Orionis,              K 

3 

4o  38.64 

2.846 

99  43  38.6 

1.66 

359 

i845 

32  Aurigae,              v 

5 

4t     5.72 

4.i56 

5o  54     6.0 

1.68 

36o 

i849 

3  1   Camelopardi, 

5 

4l     32.  08* 

5.365 

3o     9   i3.i* 

i.57 

36i 

i854 

3o  Aurigae,              £ 

5 

42  i6.53 

5.019 

34  20     9.1 

1.52 

362 

i855 

Pictoris, 

5 

42  19.03 

1.667 

i36  39   17.7 

1.4? 

363 

1861 

Pictoris,             /3 

41 

43  43.82 

i.4i6 

i4i     7  22.3 

i.54 

364 

i863 

1  36  Tauri, 

41 

43  54.o6* 

3.77i 

62  25  44.6* 

i.35 

365 

1868 

Doradus,            6 

41 

44  3o.5o 

0.077 

i55  47  35.3 

i  .07 

366 

i87i 

1  5  Leporis,             6 

5 

44    52.20 

2.578 

no  53  43.3 

0.66 

367 

i876 

54  Orionis,              x1 

5 

45  3o.o3* 

3.552 

69  45  24.7* 

i  .1-7 

368 

i878 

Columbae,           ft 

3 

45  4o.6i* 

2.  109 

125  49  43.o* 

i'.53 

36g 

i883 

58  Orionis,              a 

i 

47     3.i4* 

3.249 

82  37  33.o* 

i  .i3 

37o 

1  884 

Pictoris,             y 

41 

47    6.37 

I.  0?3 

l46    12    21.3 

i  .06 

37i 

i885 

33  Aurigae,              6 

31 

47  10.44* 

4.937 

35  44     3.4* 

I  .01 

372 

1890 

Pictoris, 

5 

47  29.53 

1.338 

142     8  4i.i 

i.  08 

373 

1891 

Columbae,           A 

5 

47  39.73 

2.  167 

123  5o  11.  8 

1.  17 

374 

i895 

34  Aurigae,              ft 

2 

48  3i.46* 

4-4o4 

45    4  27.7* 

0.97 

375 

i897 

35  Aurigae,              TT 

5 

48  48.ii* 

4.452 

44    4  59.6 

0.96 

376 

1900 

37  Aurigae,              0 

4 

49  29.59* 

4.092 

52  48  12.4* 

0.81 

377 

1901 

1  6  Leporis,              rj 

4 

49  34-47* 

+2!735 

io4  n   56.3* 

i  .06 

378 

1905 

Doradus,             e 

5 

5o     2.41 

—  O.  I2O 

i56  56   16.1 

i.  08 

379 

1922 

Columbae,           y 

4 

52  i3.o3* 

-f-2.127 

125  18   10.8* 

0.61 

38o 

1928 

6  1   Orionis,             /j. 

5 

54     7.97 

3.302 

80    21     28.O 

o.5o 

38  1 

1933 

Puppis, 

5 

54  33.  3i 

!.833 

l32    49    32.2 

o.5o 

382 

1934 

64  Orionis,             #3 

5 

54  34.  74 

3.559 

7o  18  46.  i 

o.43 

383 

i938 

i   Geminorum, 

5 

55     o.i4* 

3.648 

66  44     2.6* 

o.33 

384 

1939 

62  Orionis,              #* 

5 

55     0.76* 

3.565 

69  5i  48.i* 

o.4o 

385 

1943 

37  Camelopardi, 

5 

56  44.97* 

5.298 

3t     3  10.4* 

0.29 

386" 

igSS 

67  Orionis,              v 

41 

59     0.49* 

3.428 

75   i3     7.4* 

0.06 

387 

i959 

1  8  Leporis,             6 

41 

5g    22.12 

2.718 

io4  55  34.4 

—  o.o5 

388 

i979 

4o  Camelopardi, 

5 

6     2   11.99* 

5.393 

29  58     3.4* 

-f-     0.22 

389 

i98o 

Camelopardi, 

5 

2   i8.53* 

6.623 

20  38   i3.2* 

0.29 

39o 

I982 

Columbae,           6 

5 

2    23.  37 

2.069 

127  i4     6.9 

O  .21 

39i 

i99o 

7o  Orionis,              £ 

5 

3  24.69 

3.4i5 

75  45  44.8 

0.34 

392 

I992 

i    Lyncis, 

5 

4     4.90* 

5.538 

28  26  4o.4* 

o.37 

393 

I994 

Monocerotis, 

5 

4  32.98 

2.918 

96  3i  12.5 

o.4o 

394 

2001 

44  Aurigae,             /c 

4 

5  49.i4* 

3.828 

60  27    9.0* 

0.80 

395 

2OO2 

7  Geminorum,      77 

4 

5  49.35* 

3.624 

67  27   18.2* 

0.  52 

396 

2OO7 

2   Lyncis, 

41 

6    23.21* 

5.3IO 

3o  56  33.3* 

o.56 

397 

20l5 

5  Monocerotis, 

41 

7  32.45 

2.928 

96  i4    O'9 

o.79 

398 

2034 

Columbae,           K 

41 

n    12.95 

2.i33 

125     5  37.2* 

i  .01 

399 

2044 

46  Aurigae, 

5 

i3  20.48* 

4.629 

4o  38  33.2* 

1.23 

4oo 

2047 

1  3  Geminorum,      //        3 

6  i3  53.i3* 

+  3.636 

67  24  53.3* 

+   1-34 

TABLE   XXX. 


415 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a                  b                  c 

d 

a' 

b' 

c' 

d' 

35i 

+8.2416 

+9.1676 

+9.7086 

—  8.i899 

—0.0339 

+9.0291 

—o.383o 

+9.9968 

352 

7.89i3 

8.8212 

o.48o5 

—  6.4398 

—  9.6710 

+  7.6166 

o.3693 

9.9970 

353 

7.9563 

8.9o34 

0.3363 

—  7.  7o56 

—9.9624 

+8.7995 

0.3524 

9.9978 

354 

7.8357 

8.8563 

o.4oi3 

-7.4i85 

—9.8932 

+  8.6602 

0.2796 

9.9980 

355 

7.9ooo 

8.9323 

o.6i84 

+7.7ooi 

+9-4648 

—  8.7659 

0.2681 

9.9981 

356 

7.8o93 

8.8632 

0.5656 

+  7.4272 

+8.2672 

—8.6628 

0.2466 

9.9988 

357 

7.7755 

8.837i 

o.434o 

—7.i852 

—9.8296 

+  8.3464 

0.2890 

9.9984 

358 

7.7563 

8.8287 

o.4537 

—  6.984i 

-9.7763 

+8.1689 

0.2283 

9.9986 

359 

7-8499 

8.9325 

0.6184 

+7.6497 

+9-4649 

-8.7167 

0.2181 

9.9986 

36o 

8.0287 

9.  I2l5 

o.7295 

+7.9656 

+9.8088 

—8.8426 

0.2080 

9.9986 

36i 

7.96o6 

9.o7i3 

o.7oo9 

+7.8774 

+9.7623 

—  8.8o48 

o.  1902 

9.9987 

362 

7.8743 

8.9860 

0.219-7 

-7-7359 

—  o.oo99 

+8.  7486 

o.  1891 

9.9987 

363 

7.877o 

9.0251 

O.l5l2 

—7.7683 

—  0.0207 

+8.7421 

o.i53i 

9.9989 

364 

7.  7225 

8.8752 

o.576o 

+7.3880 

+8.8865 

—8.6117 

o.i485 

9.9989 

365 

8.o4o7 

9.2101 

9.oi45 

—  8.0007 

—  o.o366 

+8.7896 

o.i3i8 

9.9990 

366 

7.6728 

8.8525 

o.4o85 

—  7.2261 

—9.8816 

+8.37i6 

o.  1216 

9.9991 

367 

7.6524 

8.85o7 

o.55i8 

+7.1915 

-8.7917 

—  8.3399 

o.  io3i 

9.9991 

368 

7-7I07 

8.9142 

o.3237 

-7.4781 

—9.9713 

+8.563i 

0.0979 

9.9992 

369 

7.5793 

8.8268 

o.5no 

+6.6877 

—9.48o3 

—7.8601 

o.o54o 

9.9998 

37o 

7.8286 

9.o78o 

o.oSig 

-7.7482 

—  0.0298 

+8.6696 

0.0622 

9.9998 

37i 

7.8o5o 

9.o568 

0.6925 

+  7.7i45 

+9.7469 

-8.6670 

0.0498 

9.9998 

372 

7.7728 

9.o353 

o.i3i3 

—  7.6702 

—  O.O23l 

+8.6342 

0.0890 

9.9994 

373 

7.6353 

8.9039 

o.3376 

-7.38n 

—  9.9621 

+8.4766 

o.o33i 

9.9994 

374 

7.6732 

8.9733 

o.6437 

+  7.6221 

+9.5999 

—8.5483 

0.0016 

9.9996 

375 

7.67o3 

8.98io 

0.6483 

+7.6266 

+9.6187 

—8.  545i 

9.9910 

9.9996 

376 

7.5838 

8.9222 

0.6111 

+  7.3653 

+9.4120 

—8.4426 

9.9634 

9.9996 

377 

7.495i 

8.8369 

+0.4366 

—6.8848 

—9.8233 

+8.o474 

9-9599 

9.9996 

378 

7.8689 

9.2806 

—8.8248 

-7.8327 

—0.0371 

+8.6018 

9.9402 

9.9996 

379 

7.4429 

8.9n9 

+  0.3272 

—7.2047 

—  9.9693 

+8.2926 

9.  8829 

9.9998 

38o 

7.2383 

8.83oo 

o.5i82 

+6.4623 

—9.4166 

—  7.6322 

9.7104 

9-9999 

38i 

7.3343 

8.9584 

0.2629 

-7.1666 

—9.9991 

+8.2081 

9.6780 

9-9999 

382 

7.2243 

8.85oo 

o.55oi 

+6.7618 

—8.8561 

—7.9017 

9.6764 

9-9999 

383 

7.I99I 

8.86o7 

o.56i7 

+6.7967 

-7.6798 

—  7.9360 

9.64o6 

9-9999 

384 

7.i889 

8.85i2 

o.55i5 

+6.7268 

—8.8007 

-7.8745 

9.6898 

9-9999 

385 

7.2638 

9.1114 

o.7235 

+  7.i967 

+9.8012 

—8.0853 

9.4547 

o.oooo 

386 

6.4745 

8.8385 

0.5345 

+  6.8812 

—  9.2162 

—  7.0427 

8.9382 

o.oooo 

387 

+6.2787 

8.8388 

0.4336 

—6.6896 

—  9.83o5 

+6.8608 

—8.  7421 

o.oooo 

388 

—  7.io65 

9.  1254 

o.73i5 

—  7.o442 

+9.8126 

+7.9188 

+9.2884 

o.oooo 

389 

7.2796 

9.2768 

0.8209 

—  7.25o8 

+9-8937 

+  7.9740 

9.8060 

o.oooo 

39o 

6.94n 

8.9229 

0.3l27 

+6.7229 

—  9.978o 

—  7.8000 

9.8204 

o.oooo 

39i 

7.OI02 

8.8874 

0.5327 

—  6.4oio 

—  9.2423 

+  7.5635 

9-4749 

o.oooo 

392 

7.396o 

9.  i46o 

0.7434 

—7.34oi 

+9.8276 

+8.1941 

9.6622 

9-9999 

393 

7.  1245 

8.8266 

o.465i 

+6.1797 

—  9.7362 

—  7.353o 

9.6000 

9.9999 

394 

7.2892 

8.8843 

o.583o 

—6.9822 

+9.0626 

+8.0978 

9.7070 

9.9999 

395 

7.2632 

8.8583 

0.5594 

—6.8469 

—  8.243o 

+7.9886 

9.7070 

9-9999 

396 

7.5579 

9.1126 

0.7243 

—7.4912 

+9.8022 

+8.3784 

9.7478 

9.9998 

397 

7.3437 

8.8263 

o.466i 

+6.3794 

—9.7323 

—  7.6629 

9.8194 

9.9998 

398 

7.6oo5 

8.9105 

0.3289 

+7.36oi 

—9.9682 

—8.4491 

9.9917 

9.9996 

399 

7-7748 

9.0094 

0.6652 

—7.6660 

+9.6766 

+8.6449 

0.0670 

9.9998 

4oo 

—  7.64o7 

+8.8578 

+o.5594 

—  7  .2261 

—  8.2406 

+8.3665 

+0.0843 

+9.9992 

416 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.       s. 

s. 

0            /               't 

" 

4oi 

2o5l 

i   Canis  Majoris,  £ 

2* 

6   i4  33.  4i 

-}-2.  3o4 

120       0       1.6 

+   i.3o 

402 

2o6l 

2  Canis  Majoris,  j3 

2* 

16     5.79* 

2.643 

107  53     8.0* 

i.4i 

4o3 

2066 

3  Canis  Majoris, 

4 

16  38.09 

2.197 

123    21     5o.I* 

i.54 

4o4 

2090 

1  8  Geminorum,      v 

4 

20     3.35* 

3.566 

69  4i   53.3* 

1.78 

4o5 

2095 

Camelopardi, 

5^ 

20  35.  i5 

io.447 

10  17  18.9 

2.41 

4o6 

2096 

Argus,                a 

i 

20  37.46 

i.33o 

i4a  36  56.4 

i.  80 

4o7 

2109 

Canis  Majoris, 

4* 

22    36.75 

2.224 

122    29    l8.4 

i  .90 

4o8 

2126 

1  3  Monocerotis, 

5 

24  47«56 

3.247 

82  33  4i.6 

2.19 

409 

2132 

4  Canis  Majoris,   £* 

5 

25  36.53 

2.5oi 

n3  18  48.6 

2.23 

4io 

2i37 

Puppis,               Z 

5 

26     6.46 

1.432 

i4o    8  13.9 

2.5o 

4n 

2l57 

5  1  Cephei, 

5 

28  32.88* 

30.723 

2  44  38.8* 

2.58 

4l2 

2i58 

Canis  Majoris, 

5 

28  33.65 

2.093 

126     7  23.3 

2.4l 

4i3 

2159 

5o  Aurigae, 

5 

28  36.  89 

4  .  294 

47  23     5.6 

2.55 

4i4 

2160 

5  Canis  Majoris,  £a 

5 

28  46.49 

2.522 

112  5o  55.7 

2.45 

4i5 

2i63 

24  Geminorum,       y 

•4 

29     2.73* 

3.469 

73  28  39.4* 

2.56 

4i6 

2171 

7  Canis  Majoris,  vz 

5 

3o     8.5i 

2.617 

109     7  62.0 

2.66 

4i7 

2176 

Carinae, 

5 

3i  4o.n 

i.3i4 

i4a  5i    17.1 

2.78 

4i8 

2182 

55  Aurigae, 

5 

82     9.67* 

4.377 

45  20   i5.6* 

2.84 

419 

2188 

Argus,                v 

3 

33   io.35 

1.834 

i33     3  59.5 

2.83 

420 

2193 

Puppis,               V 

5 

34  38.54 

1.616 

i38     5   19.3 

3.09 

4s  i 

2194 

27  Geminorum,       e 

3 

34  42,  i3* 

3.700 

64  43  32.  7* 

3.o5 

422 

2198 

42  Camelopardi, 

5 

35   16.87 

6.302 

22    l6    19.1* 

3.07 

423 

2206 

3  1  Geminorum,       £ 

4 

36  52.23* 

3.374 

76  56  52.o* 

3.38 

424 

2209 

43  Camelopardi, 

5 

37  3o.i9* 

6.527 

20  56  48.i* 

3.27 

426 

2210 

Camelopardi, 

5 

38     6.33 

8.868 

12  5o  39.7 

3.35 

426 

22l3 

9  Canis  Majoris,  a 

i 

38  3a.  39* 

2.646 

i  06  3o  5o.6* 

4.5o 

427 

22l6 

17  Monocerotis, 

5 

3g     11.22 

3.270 

81  48   18.7 

3.4i 

428 

2222 

1  8  Monocerotis, 

5 

4o    2.48 

3.i37 

87  25  4o.3 

3.52 

429 

2223 

58  Aurigae, 

5 

4o    9.28* 

4.249 

48     2   53.5* 

3.6i 

43o 

223l 

Puppis,               x 

5 

42  i  3.  24 

2.049 

127  45   57.2 

3.57 

43i 

2237 

34  Geminorum,       6 

5 

42  53.88* 

3.963 

55  5i  49.2* 

3.78 

432 

2246 

1  3  Canis  Majoris,  K 

4 

44  i4.33* 

2.242 

122     20    22.  O* 

3.86 

433 

2248 

1  5  Lyncis, 

5 

44  i6.56* 

5.227 

3i   23   17.9* 

4.o3 

434 

2252 

Canis  Majoris, 

5 

45    25.24 

2.187 

124    II     44-6 

4.o8 

435 

2256 

Argus,                r 

4 

46  12.94 

1.490 

i4o  26  17.5 

4.i3 

436 

225g 

Carinee,              B 

5 

46  35.07 

1.279 

i43  26  54.4 

4.i3 

437 

226O 

Pictoris,              a 

4 

46  39.00 

0.611 

i5i   46  53.6 

3.79 

438 

2264 

1  4  Canis  Majoris,  B 

5 

47   i3.4i 

2.791 

101    5i    17.7 

4.i4 

439 

2267 

1  6  Canis  Majoris,  o1 

4 

47  54.73 

2.492 

Il4       O       O.I 

4.i5 

44o 

2274 

20  Canis  Majoris,  i 

4i 

49  26.89 

2.676 

106   5i  48.9 

4.28 

44  1 

2293 

21   Canis  Majoris,  e 

lift 

.52  43.89* 

2.36o 

118  46  17.4* 

4.56 

442 

2295 

Puppis,               t 

5 

52  55.64 

2.  195 

123  54  4o.5 

4.58 

443 

23o5 

43  Geminorum,       C 

4 

55  12.58* 

3.567 

69   12  52.5* 

4.8o 

444 

2309 

22  Canis  Majoris, 

3^ 

55  44.79 

S.Sgi 

117  43  23.7* 

4.84 

445 

23i8 

24  Canis  Majoris,  o2 

4 

56  45.8o 

2.507 

n3  87     1.4 

4.91 

446 

2319 

23  Cants  Majoris,  y 

4 

56  58.36 

2.718 

io5  24  56.7 

4.96 

44? 

448 

2326 

2327 

Camelopardi, 
Puppis,               C 

4i 
5 

59  11.96* 
59  17.78 

i3.i46 
1.911 

7  19     2.0* 
i32     7     8.6 

5.i4 

5.1! 

449 

2338 

63  Aurigae, 

5 

7     i   19.  84* 

4.i45 

5o  26  25.9* 

5.3o 

45o 

234o 

46  Geminorum,      r 

5 

7     i   35.20* 

4-3.83i 

5g  3o  52.i* 

4-  5.39 

TABLE    XXX. 


417 


CATALOGUE    OF    1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

bed 

a' 

V 

c' 

d? 

4oi 

—7.6890 

+8.8855 

+o.36i8 

+7.388o 

—  9.9419 

—  8.5oi6 

+o.io48 

+9.9991 

4O2 

7.6916 

8.8444 

0.4216 

+7.1789 

—  9.857o 

—8.3335 

o.i484 

9.9989 

4o3 

7.7626 

8.9010 

o.34io 

+7.5029 

—  9.9595 

—8.6008 

0.1627 

9.9989 

4o4 

7.7933 

8.85oi 

o.55i9 

—7.3336 

—8.7882 

+8.4818 

o.2437 

9.9983 

4o5 

8.5248 

9.5703 

i  .0173 

—  8.5i78 

+9.9555 

+8.9458 

o.255o 

9.9983 

4o6 

7.9942 

9.0389 

O.  1232 

+7-8943 

—  O.O232 

—  8.8537 

0.2558 

9.9982 

4oy 

7.8918 

8.8957 

0.3470 

+7.6214 

-9.9545 

—  8.7235 

0.295-7 

9-9979 

4o8 

7.8609 

8.825o 

o.5in 

—  6.9730 

—9.4794 

+8.i454 

0.3355 

9-9975 

409 

7.9082 

8.8582 

0.3976 

+7.5o56 

—9.8986 

-8.6447 

0.3495 

9.9973 

4io 

8.0727 

9.0143 

0.1702 

+7.9579 

—  0.0173 

—8.9408 

o.3579 

9.9972 

4n 

9.2387 

o.  i4o4 

1.4879 

—9.2382 

+9.9869 

+9.0944 

o.397i 

9.9966 

4l2 

8.OIH 

8.9133 

0.3227 

+7.7816 

-9.9710 

—  8.865o 

o.3967 

9.9966 

4i3 

8.o524 

8.9537 

0.6326 

—  7.883o 

+9.5470 

+8.9259 

o.3975 

9.9966 

4i4 

7.9571 

8.856o 

o.4ooo 

+7.5463 

—9.8949 

—  8.6869 

0.3999 

9.9966 

4i5 

7  .  9440 

8.8387 

0.5396 

—7.3979 

—9.1268 

+8.5557 

o.4o4o 

9.9965 

4i6 

7.9663 

8.8448 

0.4168 

+7.4818 

—9.8662 

—8.6333 

0.4200 

9.9962 

4i7 

8.1821 

9.o388 

o.  1214 

+8.o836 

—0.0218 

—  9.o4o6 

o.44i3 

9.9958 

4i8 

8.1176 

8.9676 

o.64i4 

-7.9645 

+9.588i 

+8.9926 

0.44-79 

9«9957 

419 

8.1193 

8.9557 

0.2634 

+7.9536 

—  9.9968 

—  8.9934 

o.46i3 

9.9954 

420 

8.1769 

8.9942 

o.2o36 

+8.o486 

—  O.OIIO 

—9.0495 

o.48oo 

9-995° 

421 

8.o46i 

8.8626 

0.5676 

—7.6765 

+8.4728 

+8.8089 

o.48o7 

9.995o 

422 

8.4309 

9.2401 

0.7991 

-8.3972 

+9.875i 

+9.  1  520 

o.4879 

9.9948 

423 

8.0399 

8.8296 

0.5285 

-7.3938 

—9.3008 

+8.5585 

0.5069 

9.9944 

424 

8.4826 

9.2648 

o.8i4o 

—8.4529 

+9-8844 

+9.  1822 

o.5i42 

9.9942 

425 

8.6957 

9.4709 

0.9471 

—8.6847 

+9.9370 

+9.2078 

O.52IO 

9  .  9940 

426 

8.0659 

8.836o 

0.4281 

+7.5i96 

—9.8429 

—  8.6774 

0.5259 

9.9938 

427 

8.0592 

8.8220 

o.5i33 

—  7.2i3i 

—9.4609 

+8.3848 

o.533i 

9.9936 

428 

8.o645 

8.8177 

o.4955 

—  6.7I65 

—9.5903 

+  7.8922 

0.5423 

9.9933 

429 

8.1938 

8.9458 

0.6288 

—  8.oi89 

+9.5258 

+9.0664 

0.5435 

9.9933 

43o 

8.1889 

8.9186 

O.3l22 

+7-976o 

-9.9755 

—  9.o5oo 

o.565i 

9.9926 

43  1 

8.i757 

8.8984 

0.5978 

—7.9248 

+9.2838 

+9.0188 

o.57i9 

9.9924 

432 

8.1800 

8.8890 

o.35o2 

+7.9o83 

—  9.9502 

—  9.0112 

0.5852 

9.9919 

433 

S.SgoS 

9.0991 

0.7178 

—8.3217 

+9.7866 

+9.2i45 

0.5855 

9.9919 

434 

8.2006 

8.8978 

0.3384 

+7.95o3 

—  9.959o 

—  9.o44o 

o.5965 

9.9914 

435 

8.32i5 

9.0109 

0.1717 

+8.2085 

—  o.oi36 

—  9.i887 

o.6o39 

9.9911 

436 

8.354i 

9.0400 

O.I  I  54 

+8.2589 

—  0.0192 

—  9.2IOO 

o.6o73 

9.9910 

43y 

8.4549 

9.1401 

9.7997 

+8.3999 

—0.0287 

—  9.25o7 

o.6o79 

9.9909 

438 

8.i442 

8.8240 

0.4465 

+7-4569 

—9.7964 

—8.6236 

o.6i3i 

9.9907 

439 

8.i8o3 

8.8536 

o.3959 

+7.7896 

—  9.9000 

—8.9264 

o.6i93 

9.9904 

44o 

8.1736 

8.8328 

0.4273 

+7.636i 

-9.8443 

—8.7931 

0.6328 

9.9898 

44  1 

8.2392 

8.8695 

0.3721 

+7.9216 

-9.9289 

—  9.o4o5 

o.66o3 

9.9884 

442 

8.2645 

8.8932 

o.34i6 

+8.0IU 

-9.9558 

—  9.  1062 

o.66i9 

9.9883 

443 

8.23o8 

8.84o4 

o.55i9 

-7.7808 

—  8.786o 

+8.9277 

o.6799 

9.9873 

444 

8.2586 

8.8639 

0.3781 

+7.9263 

—  9.9220 

—  9  .  o494 

o.684o 

9.9870 

445 

8.25i3 

8.8485 

0.3986 

+7.854i 

—9.8954 

—8.9922 

o.69i7 

9.9865 

446 

8..23o8 

8.8263 

0.4335 

+7.6554 

—9.83oo 

—  8.8i56 

o.6932 

9.9864 

447 

9.1262 

9.7042 

i.n85 

—9.1227 

+9.956o 

+9.4o38 

o.7o96 

9.9853 

448 

8.36i6 

8.9390 

0.2792 

+8.1882 

—9.9869 

-9.2345 

0.7I02 

9.9853 

449 

8.3592 

8.9211 

0.6166 

—8.1632 

+9-4465 

+9.2264 

o.7245 

9.9843 

45o 

—8.3i26 

+8.8726 

+o.583i 

—8.0179 

+9.0611 

+9-1294 

+0.7263 

+9.984i 

418 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.   m.        s. 

s. 

0             /                 // 

" 

45  1 

2345 

25  Canis  Majoris,  6 

3* 

7     2  17.61* 

+2.44i 

116     9  3i.i* 

+  5.36 

462 

2349 

1  8  Lyncis, 

5 

2   47.52* 

5.277 

3o     6   10.4* 

5.76 

453 

2355 

Puppis,              A 

5 

3  48.85 

2.007 

129  25     3.7 

5.56 

454 

2358 

22  Monocerotis, 

41 

4  12.27 

3.069 

90  i4  56.7 

5.56 

455 

2362 

5  1  Geminorum, 

5 

4  45.34* 

3.454 

73  35  28.9* 

5.58 

456 

2379 

Lyncis, 

5 

7    6.75 

4.58i 

4o  16  3o.5 

5.79 

45  7 

238o 

Puppis,              E 

5 

7  17.  85 

1.986 

i3o  i4  5o,4 

5.8i 

458 

238i 

64  Aurigae, 

5 

7  35.89* 

4.191 

48  5i  21.5* 

5.82 

459 

2388 

27  Canis  Majoris, 

4t 

8     8.47 

2.446 

u6     5  49.5 

5.85 

46o 

2389 

Puppis,               I 

5 

8  i6.64 

1.676 

i36  3o  46.o 

6.  20 

46  1 

2392 

Puppis,               L1 

5 

8  44.  3o 

i.  808 

i  34  55  32.o 

6.17 

462 

2398 

54  Geminorum,      A 

4t 

9  28.22* 

+3.458 

73   ii    37.7* 

5.99 

463 

2400 

Volantis,            y 

5 

10     0.28 

—0.472 

160   i5   17.0 

5.86 

464 

2407 

19  Lyncis, 

5 

10  36.  4i 

+4.926 

34  26  33.8 

6.i3 

465 

24lO 

55  Geminorum,      3 

3 

ii     9.67* 

3.597 

67  44  47.5* 

6.i4 

466 

2414 

Argus,                TT 

3 

ii  5i.o5 

2.l4l 

126  49  61.4 

6.  17 

467 

2416 

65  Aurigae, 

5 

12     0.87 

4.028 

52  57  44.4 

6.  20 

468 

2418 

3o  Canis  Majoris, 

5 

12    29.41 

2.491 

n4  4i     4.5 

6.24 

469 

2427 

Puppis,               F 

5 

i3  26.54 

2.045 

128  56  19.9 

6.3i 

4yo 

2429 

66  Aurigae, 

5 

i3  44.69 

4.175 

49     2  39.6 

6.35 

47i 

2439 

Camelopardi, 

5 

i5  12.91 

6.3i8 

21     l4    12.  I* 

6.53 

472 

2442 

60  Geminorum,       i 

4 

16  24-36* 

+3.742 

61  54  32.3* 

6.65 

473 

2447 

Volantis,             6 

5 

16  53.34 

—  O.OI2 

167  4o  57.3 

6.60 

474 

2458 

3  1  Canis  Majoris,  ij 

2 

18     9.72* 

+  2.372 

119     o  5  1  .  2.* 

6.72 

475 

2462 

3  Canis  Minoris,  p 

3 

19     0.92 

3.261 

81  24  45.3 

6.83 

476 
477 

2464 
2478 

62  Geminorum,      p 
Puppis, 

5 
5 

19  27.39* 
23  i7.73 

3.871 

2.317 

57  55  21.  i* 
121     9     i.o 

6.61 

7.18 

478 

2482 

Argus,                a 

4 

24  28.71 

i  .921 

l33       0       2.2 

7.11 

479 

2484 

Puppis, 

5 

24  52.56 

2.321 

I2O    39       4.6 

7.38 

48o 

2485 

66  Geminorum,      a2 

I* 

25     i.33* 

3.848 

57  47  17.0* 

7.34 

48  1 

2486 

68  Geminorum, 

5 

25     2.64* 

3.433 

73  5i   19.0* 

7.23 

482 

2493 

69  Geminorum,      v 

5 

26  4o.43* 

3.7i4 

62  46  33.i* 

7-49 

483 

2497 

Puppis,               nl 

4* 

27  58.35 

2.54o 

n3     9     3.9 

7.61 

484 

25oo 

Puppis,              g 

28  18.20 

2.475 

n5  47  33.7 

7.72 

485 

2522 

10  Canis  Minoris,  a 

i 

3i  26.78* 

3.145 

84  23  39.7* 

8.77 

486 

253o 

Puppis,               k1 

41 

32  40.95 

2.460 

116  27  48.3 

7.85 

487 

253i 

Puppis,               k* 

5 

32  4i.3i 

2.459 

116  27  52.8 

7-77 

488 

254o 

7  5  Geminorum,       a 

5 

33  55.8o 

3.764 

60  45  3o.8 

8.23 

489 

2542 

26  Monocerotis,      y 

41 

34    4.89 

2.868 

99   12   17.5 

8.02 

490 

255i 

77  Geminorum,      AC 

4 

35  23.  i5* 

3.634 

65   i4  49.5* 

8.16 

491 

2555 

78  Geminorum,      ft 

2 

36     7.80* 

3.682 

61   36  59.0* 

8.23 

492 

2562 

3  Puppis, 

5 

37  47.27* 

2.409 

118  35  55.2* 

8.27 

493 

257O 

Puppis,              W 

41 

38  35.45 

2.039 

i3o  34   i3.5 

8.68 

494 

258o 

Puppis,              c 

5 

39  54.63 

2.132 

127  36  23.7 

8.4o 

495 

2590 

Camelopardi, 

& 

4o  56.  7i 

9.819 

10     7  24.3 

8.65 

496 

25g4 

Puppis,              o 

5 

4l    5l.22 

2.502 

ii5  34     5  7 

8.69 

497 

2602 

Argus,               £ 

3* 

42  59.28* 

+  2.527 

ii4  29  12.5* 

8.70 

498 

26o7 

Volantis,             C 

5 

43  38.25 

—0.669 

162  i5     8.4 

9.71 

499 

261-7 

83  Geminorum,      <j> 

5 

44  18.61* 

+3.688 

62  5i     3.6* 

8.85 

5oo 

2620 

Puppis,               P 

41 

7  44  40.19 

+  1.826 

i35  59  53.8 

+  8.90 

TABLE   XXX. 


419 


CATALOGUE   OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

b' 

c' 

d' 

45  1 

—8.2998 

+8.8546 

+0.3870 

+  7.9440 

—9.9107 

—  9.0732 

+0,7311 

+9.9838 

45a 

8.5558 

9.  1071 

o.7235 

—8,4929 

+9.7890 

+9.3693 

0,7345 

9.9835 

453 

8.375i 

8.9189 

o.3o4i 

+8.1779 

—9.9758 

—9.2419 

o.74i3 

9.9829 

454 

8.2656 

8.8066 

0-4864 

+  5.9040 

—9.6417 

—7.0801 

0.7439 

9.9827 

455 

8.2873 

8.8244 

o.5377 

-7.7383 

—9.1614 

+8.8963 

o.7475 

9.9824 

* 

, 

456 

8.4738 

8.9945 

0.6610 

-8.3563 

+9.6532 

+9-3429 

0.7626 

9.9811 

457 

8.4028 

8.9222 

0.2983 

+8,2i3i 

—  9-9775 

—9.2719 

0.7638 

9.9810 

458 

8.4io5 

8.9279 

0.6220 

—8.  2^287 

+9.4812 

+9.2816 

0.7657 

9.9808 

459 

8.3374 

8.85n 

0.3882 

+7.9808 

-9.9087 

—  9.  i  i  02 

0.7690 

9.9805 

46o 

8.4539 

8.9666 

0.2363 

+8.3i45 

—  9,9966 

—9.3283 

0.7699 

9.9804 

46i 

8.4444 

8.9540 

0.2545 

+8.2933 

—9.9921 

—9.3194 

0.7727 

9.9802 

462 

8.3i78 

8.8226 

-j-o.5386 

—7.7790 

-9,i443 

+8.9361 

0.7772 

9.9797 

463 

8.7734 

9.2746 

—  9.6829 

+8.7471 

—  0.0188 

—  9.4519 

0.7804 

9.9794 

464 

8.5532 

9.o5o5 

-{-0.6927 

—  8.4695 

+9,7328 

+9.3981 

o.784o 

9.9791 

465 

8.3426 

8.8363 

0.5553 

—7.9209 

—8.6128 

+9.0634 

o.7873 

9.9787 

466 

•8.4097 

8.8989 

0.3260 

+8.i875 

—  9.9626 

—  9.2669 

o.79i3 

9.9783 

467 

8.4119 

8.9000 

o.6o53 

—8,1917 

+9.355i 

+9.2699 

o.7923 

9.9782 

468 

8.3584 

8.8434 

0.3956 

+7.9792 

—9.8983 

—9.  1  1  36 

o.795i 

9.9779 

469 

8.43i3 

8.9103 

o.3io8 

+8.2296 

—9.9704 

—  9.2966 

o.8oo5 

9.9773 

4yo 

8.4459 

8.9230 

o.62o3 

—8,2624 

+9-4685 

+9.3i66 

0.8023 

9.9771 

4?z 

8.7732 

9.2411 

0.8012 

—8.7427 

+9.8604 

+9.4778 

o.8io5 

9.9762 

472 

8.3932 

8.8537 

+0.5734 

—8,  066  1 

+8.7938 

+9.1878 

0.8171 

9.9754 

473 

8.7619 

9.2195 

—  7.6812 

+8.7281 

—  o.oi63 

-9.4837 

0.8197 

9-975i 

4?4 

8.4o65 

8.8564 

+o.375i 

+8,0923 

—  9.9225 

—  9.2101 

0.8266 

9.9742 

4?5 

8.3577 

8.8025 

o,5i33 

—  7.53i8 

—  9.46o3 

+8.7030 

o.83u 

9-9737 

476 

8.4271 

8.8692 

0.5864 

—  8.1522 

+9.1199 

+9.2564 

0.8334 

9.9734 

477 

8.4424 

8.8622 

o,3646 

+8.i56i 

—9.9319 

—9.2646 

o.853i 

9.9707 

478 

8.5i65 

8.9296 

o.28o5 

+8.35o3 

—9.9794 

—9.3905 

0.8589 

9.9698 

479 

8.4479 

8.8588 

o.3677 

+8.i553 

—9.9287 

—  9.2660 

0.8609 

9.9695 

48o 

8.4558 

8.8659 

o.586i 

—8.1826 

+9.1126 

+9.2861 

0.8616 

9.9694 

48  1 

8.4009 

8.  8108 

0.5355 

—  7.845o 

—  9.  1976 

+9.oo36 

0.8617 

9.9694 

482 

8.4422 

8.843i 

0.5694 

—8.1026 

+8.5944 

+9.2277 

0.8695 

9.9682 

483 

8.4338 

8.8276 

0.4049 

+8.0284 

—9.8829 

—9.1680 

0.8757 

9.9672 

484 

8.4445 

8.8364 

o.SgSo 

+8.o83i 

—9.8998 

—9.2136 

0.8772 

9.9669 

485 

8.4i54 

8.7905 

o.5o4o 

—  7.4o52 

—9.5339 

+8.5792 

0.8916 

9.9645 

486 

8.4669 

8.8355 

0.3907 

+8.u59 

—  9.9021 

—9.2439 

0.8971 

9.9635 

487 

8.4669 

8.8355 

0.3907 

+8.1159 

—  9.9021 

—9.2439 

0.8972 

9.9635 

488 

8.4835 

8.8456 

0.5749 

—8.1724 

+8.84o7 

+9.2893 

0.9026 

9.9625 

489 

8.43o6 

8.7919 

0.4582 

+7.6346 

-9.7597 

—  8.8o5i 

0.9033 

9.9623 

490 

8.4725 

8.8270 

o.56o4 

—8.0944 

—8.0453 

+9.2286 

0.9089 

9.9613 

491 

8.4894 

8.84o2 

o.57i8 

—8.1665 

+8.7i93 

+9.2869 

0.9121 

9.9606 

492 

8.4973 

8.8396 

o.38i5 

+8.1773 

—9.9121 

—9.2969 

0.9191 

9.9592 

4g3 

8.5635 

8.9018 

0.3075 

+8.3767 

—  9.964i 

—9.4334 

0.9224 

9.9585 

4g4 

8.55o6 

8.8824 

0.3298 

+8.336i 

—  9.9528 

—9.4111 

0.9278 

9-9574 

495 

9.2087 

9.5354 

0.9932 

—  9.2019 

+9.9107 

+9.6230 

0.9320 

9-9564 

496 

8.5o2i 

8.8243 

o  3967 

+8.i37i 

—9.8928 

—9.2685 

o.9356 

9.9556 

497 

8.5027 

8.8194 

+o.4oi8 

+  8.1202 

—9.  8855 

—9.  2554 

0.9401 

9.9546 

498 

8.98o3 

9.2939 

-9.  8368 

+  8.9591 

—  9.9952 

—9.6193 

0.9427 

9.9540 

499 

8.5i77 

8.8280 

+0.5666 

—8.1769 

+8.3766 

+9.3023 

0.9453 

9.9534 

5oo 

—8.6266 

4-8.9352 

+0.2620 

+8.4835 

—  9.9764 

—  9.5oi4 

+0.9467 

+9.9530 

420 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation, 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

A.     TO.            *. 

s. 

"~     o       ~'           7i 

5oi 

2622 

9  Puppis, 

5 

7  44  49.57 

+2.780 

io3  3o  10.8 

+    9.19 

5O2 

2629 

Puppis, 

5 

46  4o.65 

2.263 

124  19  58.  o 

8.91 

5o3 

2634 

Puppis,               a 

5 

47     3.67 

2.o63 

l3o    II     32.2 

9.  12 

5o4 

2-635 

Fuppis,              d 

5 

47    20,4l 

2.132 

128  28  38.4 

0.  12 

5o5 

2642 

Velorum, 

5 

48  49.76 

1.678 

139  i3  3i.2 

9.35 

5o6 

2644 

Puppis,              R 

4 

48  53.  3i 

1.737 

137  42  45.i 

8.86 

607 

2665 

Argus,               x 

4 

52    67.66 

1.618 

142  34  54.4 

9.60 

5o8 

2666 

Puppis, 

5 

53     8,56 

2.688 

107  69  26.7 

9.60 

5oo 

2670 

Velorum, 

5 

53  56.  71 

1.766 

i38  5o  20.2 

9-48 

5io 

2673 

Canis  Minoris, 

5 

54  27.69 

3.129 

87  i5  26.9 

9.49 

5n 

2697 

27  Lyncis, 

5 

57     9.06* 

4.556 

38    4    0.2* 

9.82 

5X2 

2707 

55  Camelopaxdi, 

5 

67  49»n* 

6.096 

21     5  3i.o* 

9.86 

5i3 

2710 

Argus,               £ 

2i 

58  18.98 

2.114 

129  35     0.9 

9.96 

5i4 

2714 

10  Cancri,              //2 

5 

58  55.89 

3.545 

67  59     7.9 

9-97 

5i5 

2728 

Argus,               p 

3* 

8     i     9.42* 

2.558 

n3  62  3o,3* 

io.o4 

5i6 

2730 

1  4  Cancri,              i>* 

4 

r  24,66 

3.63o 

64    2  28.9 

10.47 

617 

2736 

1  6  Puppis, 

5 

2    19.91 

2  .  682 

108  48  30.7 

10.17 

5i8 

2754 

Velorum, 

5 

4  62.10 

1.849 

i36  54  17.1 

10.39 

5z9 

2755 

Argus,             -•£  : 

2 

4  54.  5o 

i  .  84o 

i36  53  47.9 

10.46 

5ao 

2769 

20  Puppis, 

5 

6  26.41 

2.761 

106  20  24.2 

10.54 

621 

2773 

Volarrtis,            e 

5 

7  26.99 

0.261 

168  10  37.9 

10.62 

622 

2774 

Puppis,              r 

.5 

7  60.06 

2.268 

126  26  53.5* 

10.71 

523 

2776 

3o  Lyncis, 

5 

8  17.16* 

4.907 

3i  47  42.5* 

io.63 

624 

2778 

17  Cancri,              /? 

4 

8  22.72 

3.263 

80    21     22.2 

10.72 

525 

2792 

Lyncis, 

5 

12    25.34* 

4.692 

36  18     8.6* 

10.99 

526 

2793 

3  1  Lyncis, 

5 

12    32.94* 

4-i4i 

46  20     8.2* 

ii  .06 

527 

2795 

Puppis,              q 

5 

12    66.69 

2.244 

126  ii   5  i.  i* 

10.99 

528 

2802 

Puppis,              w 

5 

i5  29.23 

2.377 

122  34  60.1 

11.26 

520 

2819 

i  Ursae  Majoris,  o 

4 

17  45.46* 

6.066 

28  4y  10.8* 

n.46 

53o 

2823 

Velorum,           B 

5 

17  55.2O 

i.83« 

i  38     o  4o.5 

n.3o 

53i 

2832 

Argus,               e 

2 

19  26.89 

1.241 

149     i  4o.3 

11.35 

532 

2842 

2  Ursae  Majoris,   A 

5 

21        6.  2O 

+6.476 

24  20  69.0 

11.66 

533 

2849 

Chamseleontis,  a 

41 

22     18.70 

—1.426 

166  26  38.7 

ii.  69 

534 

2856 

Volantis,            17 

5 

23    22.60 

—0.480 

162  54  53.2 

11.80 

535 

2863 

Volantis,            /? 

5 

24    5.4i 

+0.662 

166  38  i3.6 

I2.OO 

536 

2870 

Chamaeleontis,  0 

5 

26     2.81 

—1.662 

166  59  53.1 

ii.  81 

537 

2884 

4  Ursae  Majoris,    n 

5 

27    2.92* 

+5.358 

26     9  i5.6* 

11.99 

538 

2901 

4  Hydrse,              -6 

4 

29  4'2.64* 

3.i84 

83  46  36.i* 

12.  17 

539 

2911 

5  Hydree,              a 

5 

3o  66.07 

3.i47 

86     8     6.9 

12  .29 

54o 

2926 

Velorum,           e 

5 

32    22.28 

2.107 

i32  27  67.7 

12.35 

54i 

2935 

Mali,                  b 

5 

34  i4.i3* 

2.363 

124  46  60.2* 

12.  60 

542 

2937 

43  Cancri,              y 

4* 

34  35.91* 

3.488 

67  69  44-7* 

12.60 

543 

2945 

7  Hydras,              77 

5 

35    23.02 

3.i43 

86     3  58.i 

12.69 

544 

2947 

Velorum,           b 

5 

35  39.07 

1.989 

i36     7     3.3 

12.67 

545 

2950 

Argus,               o 

4 

35  69.80 

1.717 

l42    23    29.8 

12.69 

546 

2953 

47  Cancri,               6 

41 

36     9.26* 

3.425 

71   17  53.o* 

12.87 

547 

2962 

Cannes,              d 

5 

37  18.24 

1.344 

149   i3  35.1 

12.55 

548 

2964 

Mali,                  a 

41 

37  34-i3* 

2.4lO 

122  38  55.4* 

12.64 

549 

2965 

48  Cancri,               i 

5 

37  36.76 

3.654 

60  4i  44.o 

12.80 

55o 

2971 

ii   Hydrae,               e 

4 

8  38  49.78*4-3.189 

83     2     3.5* 

+12.84 

TABLE   XXX. 


421 


CATALOGUE    OF   1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a                  b 

c 

d 

a' 

b' 

c' 

d' 

5oi 

—8.48II 

+8.7890 

-1-0.4444 

+7-8494 

—  9.8oo4 

—  9.oi33 

+  0.9473 

+9.9529 

5o2 

8.559i 

8.8582 

o.353i 

-|-8.3io4 

—  9.9357 

—  9.4o34 

o.9543 

9.95l2 

5o3 

8.5944 

8.8917 

o.3i43 

-j-8.4o42 

—  9.9579 

—  9.4633 

o.9558 

9.95o8 

5o4 

8.5848 

8.8807 

0.3268 

+8.3787 

—  9.95i8 

—  9.4485 

o.9568 

9-95o5 

5o5 

8.6691 

8.9580 

•0.2284 

+8.5483 

—  9.9800, 

—  9.5394 

o.9623 

9-9491 

5o6 

8.6563 

8.945o 

0.2464 

+8.5254 

—9.9776 

—  9  .  5294 

o.9626 

9.9490 

5oy 

8.7i53 

8.9852 

o.i85o 

+8.6i52 

—  9.9838 

—  9.575o 

0.0,772 

9.945o 

5o8 

8.52i3 

8.7905 

0.4295 

+8.0IH 

-9.  8355 

—  9.i654 

0-9779 

9-9448 

609 

8.684o 

8.9495 

0.2371 

+8.56o8 

—  9.9765 

—  9.5552 

o.98o7 

9.9440 

5io 

8.5o47 

8.7678 

o.495i 

-7.i846 

—  9.5927 

+8.3602 

o.9825 

9.9434 

5n 

8.7234 

8.9745 

o.6589 

—  8.6i95 

+  0..6223 

+0.5856 

o.99i7 

9.9406 

5l2 

8.9595 

9.2077 

0.7846 

—  8.9294 

+9.8i7o 

+  9.66i6 

o.9939 

9.9399 

5i3 

8.63o4 

8.8764 

0.324l 

+8.4347 

-9.9487 

—9-4977 

o.9956 

9.9394 

5i4 

8.5522 

8.7955 

o.549o 

—8.1261 

—8.8854 

+9.2693 

o-9977 

9.9387 

5i5 

8.5655 

8.799o 

0.4082 

+8.1727 

—  9.8729 

—  9.3o99 

i.ooSo 

9.9363 

5:6 

8.5737 

8.  8061 

o.56o2 

—  8.2149 

—  8.0969 

+9-3448 

.oo58 

9.9360 

617 

8.5543 

8.7827 

o.4279 

+8.0627 

—9.8378 

—  9  .2i5o 

.0088 

9.935o 

5i8 

8.7039 

8.92i4 

0.2668 

+8.5674 

—9.9643 

-9.578o 

.0168 

9.9321 

5i9 

8.7040 

8.92i3 

o.2669 

+8.5674 

—9.9642 

-9.578i 

.0160, 

9  .9320 

52O 

8.559i 

8.7699 

o.44o6 

+  7.0.816 

—9.8088 

—  9.i4i9 

.O2I7 

9.93o3 

5si 

8.9762 

9.i828 

9,3705 

+8.9439 

—9.9776 

—  9.  69O2 

.O247 

9.929i 

622 

8.6367 

8.84i6 

0.3547 

+8,.4ooi 

—9.9272 

—  9.487i 

.O259 

9.9286 

523 

8.8273 

9.o3o3 

o.69oo 

—8.7567 

+9.6916 

+0.6545 

.027^ 

9.0,281 

624 

8.5555 

8.7581 

o.5i36 

—  7-7795 

—  9.4562 

+8.9494 

.027* 

9.928o 

525 

8.7889 

8.9746 

0.6623 

—  8.6952 

+9.6214 

+9.6436 

.o396 

9.923i 

526 

8.7023 

8.8875 

o,6i69 

—  8.54i4 

+  9.42I9 

+  9.5768 

.o399 

9.9229 

527 

8.6559 

8.8395 

0.3526 

+8.4272 

-9.9265 

—  9.  5ioi 

.o4n 

9.  9224 

528 

8  .  6444 

8.8i75 

0.3731 

+8.3756 

—  9.9ioo 

—  0..4773 

.0483 

9.9I92 

620 

8.8938 

9.0576 

0.7057 

-8.8365 

+9.7io8 

+  9.695s 

.0547 

9.9i63 

53o 

8,75i4 

8.9i46 

0-2663 

+8.6226 

—9.9557 

—  9.624i 

.o55i 

9.9161 

53i 

8.8695 

9.0265 

o.o94^ 

+8.8027 

—  9.  9.670 

—  9.69O2 

.  o59c 

9-9i4i 

532 

8.97o3 

9.1206 

+o.739o 

—  8.9298 

+9.752o 

+  9-72U 

.o638 

9.9n9 

533 

9.2188 

9.3642 

—  o.i582 

+9.2065 

—  9.95o6 

-9.7525 

.0670 

9.9io3 

534 

9.1235 

9.2647 

—  9.6586 

+9.1039 

-9.9554 

—  9.748o 

.o698 

9.9o88 

535 

8-9779 

9.  n62 

+0.8342 

+8.9374 

—  9.96i8 

—  9.7290 

.0717 

9.9o78 

536 

9.2437 

9.3783 

—  O.2O37 

+9.2324 

—  9.946i 

—9.7607 

.0742 

9.9o65 

537 

8.9725 

9.0992 

+0.7285 

—8.9293 

+9.733o 

+9.7338 

.o793 

9-9o37 

538 

8.6102 

8.7264 

o.5o33 

—7.6453 

—  9.5382 

+8.8188 

.0860 

9-8999 

539 

8.6116 

8.723o 

0.4972 

—  7.44o3 

—  9.5794 

+8.6i54 

.o889 

9.898i 

54o 

8.  7463 

8.8520 

0.3238 

+8.5757 

—  9.93o2 

—  9.6196 

.O92f 

9.896o 

54i 

8.7041 

8.8026 

0.3701 

+8.46o3 

—  9.9o45 

—  9.  55o9 

.o969 

9.8932 

542 

8.6523 

8  .  7494 

0.5432 

—  8.2260 

—  9.o4o2 

+9.3692 

.0978 

9.8926 

543 

8.6224 

8,7164 

o.4972 

-7.4587 

—  9.5792 

+  8.6338 

.0996 

9.89i5 

544 

8.7811 

8.874i 

o.2986 

+8.6389 

-9.9353 

—  9.6558 

.ioo3 

9.89io 

545 

8.8373 

8.9289 

o.236o 

+8.736i 

—  9.9447 

—  9.6977 

.IOII 

9.8905 

546 

8.6467 

8.7377 

o.5343 

—  8.  1527 

—  9.2093 

+9.3o53 

.ioi5 

9.89o3 

547 

8.9169 

9.oo34 

O.I25l 

+8.85o9 

—9-9477 

-9.736o 

.io4i 

9.  8885 

548 

8.7011 

8.7867 

o.38i8 

+8.433i 

—  9.8934 

-0.5345 

-1047 

9.8881 

549 

8.6860 

8.7714 

0.5624 

-8.3757 

—6.3oio 

+  9-4923 

.io48 

9.8880 

55o 

-8.6325 

+8.7i32 

+o.5o47 

-7.7i63 

—9.5283 

+8.8892 

+   .1076 

+9.8861 

422 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.       s. 

s. 

o        '          " 

» 

55i 

2978 

1  3  Hydras,               p 

5 

8  4o  29.n 

-}-3.  186 

83  36  39.5 

-J-I2.O.5 

552 

2979 

Argus,                6 

3 

4o  33.66 

1.652 

i44     9  38.6 

1  3.  06 

553 

298l 

Velorum,            a 

5 

4o  56.  75 

2.034 

i35  29  43.6 

12.  99 

554 

2998 

Carinae,             / 

5 

42  49.93 

-f-i.562 

i46  i  3     9.1 

12.83 

555 

3o23 

Chamaeleontis,  rj 

5 

46  i7.84 

—  1.816 

i  68  24  53.5 

12.82 

556 

3o32 

1  6  Hydra,              C 

4 

4?  27.9o 

+3.i84 

83  29   n.  2 

i3.38 

557 

3o48 

9  Ursae  Majoris,    * 

3* 

48  54.53* 

4.125 

4l     22     24.4* 

18.76 

558 

3o49 

8  Ursae  Majoris,   p 

5 

48  56.  3o 

5.553 

21  4y  29.9 

i3.49 

559 

3o55 

65  Cancri,                a 

4 

5o  16.71* 

3.293 

77  33  53.8* 

1  3.  60 

56o 

3o59 

Lyncis, 

4 

5o  53.o7* 

3.93i 

47  37  37.7* 

i3.89 

56i 

3o73 

Carinae,              bl 

4 

53  18.01 

i.458 

i48  39     4.8 

i3.63 

562 

3o75 

12  Ursae  Majoris,    AC 

4 

53  21.54* 

4.i4o 

42  i5   i5.6* 

i3.88 

563 

3o87 

1  1  Ursae  Majoris,    ol 

5 

55     9.18* 

5.399 

22     3l     48.0* 

i3.92 

564 

3o89 

Carinae,              bz 

4 

55  43.ii 

i.46i 

i48  3o  37.5 

i3.4o 

565 

3o97 

Lyncis, 

5 

56  58.42* 

3.855 

5o  57     7.0* 

i4.o5 

566 

3o99 

1  3  Ursae  Majoris,    oz 

5 

57     7.36* 

5.4o7 

22    I  5    44-O* 

i4.n 

567 

3io6 

1  5  Ursae  Majoris,  / 

5 

58  i5.49 

4.287 

37  47  37.9 

14.12 

568 

3io8 

1  4  Ursae  Majoris,    r 

5 

58  29.38 

5.o44 

25     52     52.  7 

14.18 

569 

3no 

Velorum,           c 

5 

58  59.47 

2.084 

i36  3o   1  1.  6 

l4«2O 

670 

3m 

•76  Cancri,              K 

5 

59  37.i2* 

3.262 

78  43  62.8* 

i4.i3 

57i 

3n4 

Volantis,             a 

# 

9     o    4.02 

o.962 

i55  47  54.3 

i4.3i 

572 

3i25 

1  6  Ursae  Majoris,    c 

5 

2    26.16* 

4.844 

27  57  5i.o* 

14.39 

573 

3i26 

Argils,    '            "k 

3 

2    29.o5 

2.  2O2 

i32  49  45.  7 

i4.4i 

574 

3i36 

Carinae,               G 

5 

4    43.28 

O.2OO 

162     o     6.6 

14.99 

575 

3i4o 

1  8  Ursae  Majoris,    e 

5 

5  21.45* 

4.382 

35  21  46.7* 

14.49 

576 

3i46 

22  Hydrae,               6 

41 

6  33.53 

3.i3i 

87     3  19.9 

14*90 

577 

3i49 

Carina&,               a 

5 

7     i'i?' 

1.562 

i  48  21   19.5 

14.97 

578 

3i52 

Carinae,              i 

5 

7    52.21 

1.354 

i5i  42     9.9 

14.63 

579 

3i62 

38  Lyncis, 

4 

9  29.8i 

3.763 

52   33   55.6 

i4.8o 

58o 

3i63 

Velorum,           / 

5 

9  42.55 

2.37I 

127  56  44«4 

14.57 

58i 

3i77 

Argus,                /3 

i 

II  32.  18 

o.69o 

i5g     6     0.6 

i4.79 

582 

3i78 

4&  Lyncis,               a 

4 

II     54.  29 

3.682 

54  58   37.2 

14.92 

583 
584 

3i86 
3i87 

Argus,                i 
Velorum,            K 

2 

5 

i3     4.63 
i3     6.71 

1.602 
2.006 

i  48   38  49.6 

l4o    25     23.2 

14.90 
i5.o2 

585 

3i95 

Mali,                  h 

5 

i4  5i.55 

2.663 

n5   19  45.6* 

14.96 

586 

3i99 

Draconis, 

5 

i5  i5.42* 

9.258 

8     i     6.5* 

i5.i3 

587 

3204 

i  Leonis,              « 

5 

i5  54.6o 

3.5i4 

63   10  28.2 

i5.i6 

588 

32i3 

Argus,               K 

3 

i7  28.  39 

i.85i 

l44    22     2O.  2 

15.29 

589 

3221 

23  Ursae  Majoris,    h 

4 

i9  38.53* 

4.833 

26     1  7     12.0* 

i5.3o 

590 

3223 

3o  Hydrae,              a 

2 

20  12.94* 

2.951 

98       0    40.2* 

i5.35 

59i 

3226 

Hydra, 

5 

20    20.77 

2.989 

95         2:5               7.7 

i5.38 

59s 

3232 

24  Ursae  Majoris,    d 

5 

21       7.77* 

5.482 

19   3o  54.9* 

i5.4i 

593 

3242 

25  Ursae  Majoris,   6 

3 

22  47-56* 

4.049 

37   38   33.i* 

16.12 

594 

3246 

4  Leonis,               A 

4* 

23       0.26* 

3.44i 

66  22  24.9* 

i5.58- 

595 

3249 

Carinae,              n 

5 

23    29.12 

i.3i8 

i54  16  5i»9 

i5.58 

596 

325o 

5  Leonis,              I 

5 

23  5i.37* 

3.245 

78     2   i9.3* 

i5.64 

597 

3257 

Argus,                i}> 

4 

24  48.  i5 

2.367 

i29  48  43.  9 

i5.54 

598 

326i 

10  Leonis  Minoris, 

5 

25        1.27 

3.707 

52  56  20.7 

i5.64 

599 

3269 

Velorum,            N 

5 

26  4o.23 

1.825 

i46  22  25.4 

15.58 

600 

3289 

Carinae,               h 

5 

9  38     5.38 

-fi.  7io 

i48  33  43.2 

+  15.91 

TABLE  XXX. 


423 


CATALOGUE  OF  1500  STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

b' 

c'                d' 

55x 

—8.6358 

+8.7ioi 

+o.5o3i 

—  7.6822 

—  9.5393 

+8.8556 

+  i.in4 

+9.8835 

552 

8.8657 

8.9397 

O.2l89 

+8.7745 

—  9.94i4 

—  9.7l82 

.1116 

9.8834 

553 

8.7884 

8.8610 

o.3o8o 

+8.64i6 

—  9.9287 

—  9.6634 

.1124 

9.8827 

554 

8.8932 

8.9585 

4-o.i9i9 

-|-8.8i29 

—  9.94O2 

—9.7341 

.1166 

9-8797 

555 

9.343o 

9.395i 

—  o.2573 

+9.334i 

—  9.9i38 

—  9.8i3o 

.1241 

9.874o 

556 

8.65n 

8.6987 

+o.5o3o 

—  7.7o59 

—  9.54oo 

+8.879i 

.1266 

9.872O 

557 

8.83i2 

8.8732 

O.6227 

—  8.7o65 

+9.4a96 

+9-7027 

.I297 

9.8695 

558 

9.o8i7 

9.i237 

o  .  744o 

—  9.o495 

+  9.72!3 

+9-7593 

.I297 

9.8695 

559 

8.6645 

8.7oi4 

o.5i69 

—  7.9976 

—  9.4236 

+9.i634 

.1325 

9.867i 

56o 

8.7869 

8.82i5 

o.5984 

—  8.6i55 

+9.24i8 

+9.66oi 

.i337 

9.8661 

56i 

8.944i 

8.9695 

o.i685 

+8.8756 

-9'9279 

—9-7679 

.i386 

9.86i8 

563 

8.8328 

8.858o 

o.6i7i 

—  8.7O2I 

+9.39i8 

+9-7°59 

.1388 

9.86i7 

563 

9.o8o6 

9.o99o 

O.7322 

—  9.o462 

+9  -60.0.4 

+9.8o56 

.1423 

9.8585 

564 

8.9472 

8.9634 

o.i757 

+8.878o 

—9-9247 

—  9.7720 

.1434 

9.8575 

565 

8.7774 

8.7888 

0.5852 

—  8.5767 

+9.o58i 

+9.643o 

.i459 

9.855i 

566 

9.o894 

9.ioo3 

o.733i 

—  9.o558 

+9.6974 

+9.8io3 

.1462 

9.8549 

567 

8.8827 

8.8893 

0.6334 

—  8.78o5 

+9-4729 

+9.7439 

.i484 

9.8528 

568 

9.o3o5 

9.o362 

o.7oi7 

—  8.9846 

+9.6542 

+9.8oo7 

.i488 

9.8523 

569 

8.8337 

8.8375 

o.3i6o 

+8.6943 

—  9.9io6 

—  9.7o82 

.i498 

9.85i4 

57o 

8.6811 

8.6826 

o.5i3i 

—  7.972I 

—  9.4582 

+9.i398 

.  i5io 

9.85o2 

57i 

9.o6o8 

9.o6o5 

9.9857 

+9.o2o8 

—  9.9l52 

—  9.8o97 

.i5i8 

9.8494 

572 

9.oo69 

8.9976 

0.6842 

—  8.953o 

+9.6i74 

+9.8ooi 

.i563 

9.8448 

573 

8.8i27 

8.8o33 

0.3432 

4-8.645x 

—9-8997 

—  9.6865 

.i564 

9.8447 

574 

9.  I922 

9.1743 

9.3448 

+9.i7o4 

—  9.899i 

—  9.8365 

.i6o5 

9.84o4 

575 

8.92o8 

8.9oo5 

o.64o6 

—8.8322 

+9'4939 

+9-77°8 

.1616 

9.839i 

576 

8.6860 

8.6612 

o.4939 

-7.3967 

—  9.6ooo 

+8.6728 

.1638 

9.8367 

577 

8.9664 

8.9398 

o.  i998 

-f-8.8965 

—  9.9o9o 

—  9.7925 

.i646 

9.8358 

578 

9.OI2O 

8.982i 

o.i386 

+8.9567 

-9.9o67 

—  9.8o86 

.1661 

9.834i 

579 

8.79o8 

8.7548 

o.5756 

—  8.5746 

+8.8261 

+9.65o6 

.  i69o 

9.83o7 

58o 

8.7942 

8.7573 

o.3739 

+8.583o 

-9.8798 

—  9.6559 

.i693 

9.83o3 

58i 

9.i4i9 

9.o98i 

9.8588 

4-9.II23 

—  9.8928 

—  9.84o7 

.I725 

9.8265 

582 

8.78i6 

8.7364 

o.568o 

—  8.54o4 

+8.4594 

+9.6298 

.i73i 

9.  8257 

583 

8.98o6 

8.93o9 

0.2068 

+  8.0.120 

—  9.9ooo 

—  9.8o44 

.I75i 

9.8232 

584 

8.8927 

8.8429 

o.2998 

+6.7796 

—  9.8974 

—  9-7599 

.I752 

9.8232 

585 

8.7437 

8.6872 

0.4238 

+8.375o 

—  9.829o 

—  9.5o7i 

.i78i 

9.8i94 

586 

9.5559 

9-4979 

o.9695 

-9.55i7 

+9-7720 

+9.8723 

.i788 

9.8i86 

587 

8.75io 

8.69o5 

o.5459 

—  8.4o54 

-8.9595 

+9.532i 

.i799 

9.8i7i 

588 

8.9388 

8.8723 

0.2685 

+8.8488 

—  9.8933 

—  9.7902 

.1824 

9.8i37 

589 

9.o6i3 

8.9865 

0.6825 

—  9.oi39 

+9.5857 

+9.8363 

.i859 

9.8o89 

59o 

8.7i28 

8.6358 

o.4698 

+7.8569 

-9.7i53 

—  9.0287 

.1868 

9.8o76 

59i 

8.7io7 

8.633i 

0.4756 

+7.6858 

—  9.6925 

—  8.8599 

.i87o 

9.8o73 

592 

9.i86i 

9.io56 

o.7389 

—  9.i6o4 

+9.66o2 

+9.86o4 

.i883 

9.8o55 

593 

8.9267 

8.8397 

0.6200 

—8.8253 

+9-374o 

+9-7873 

.I9o9 

9.8oi7 

594 

8.75u 

8.6628 

0.5366 

—8.3540 

—  9.i59o 

+9.4921 

.1914 

9.8oo8 

595 

9.0762 

8.9865 

O.I2O2 

+9.o3o9 

-9.879o 

-9-8444 

.I9I9 

9.8obi 

596 

8.7237 

8.6326 

o.5n8 

—  8.o4o3 

—  9.4684 

+9.2o68 

.I925 

9-7992 

597 

8.83o2 

8.7355 

o.3752 

+8.6366 

—  9.866o 

—  9.698i 

.i94o 

9-7970 

598 

8.8i4o 

8.7i84 

0.5685 

—  8.594o 

+8.49oo 

+  9.672I 

.i943 

9.7965 

599 

8.975i 

8.873i 

0.2610 

+8.8956 

—  9.8789 

—  9.8i5o 

.i968 

9.7926 

600 

—  9.oo62 

+8.8008 

+o.24o5 

+8.9372 

-9.8724 

-9.83o7 

+   .2018 

+9.7842 

424 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag 

Right  Ascension 
Jan.  1,  1850. 

Annual 
Variation 

North  Polar  Dist. 
Jan.  1,  1850. 

Annual 
Variation. 

h,   m.        $. 

8. 

Of                  „ 

>/ 

60  1 

33oo 

Velorum,           M 

4 

9  3i  27.89 

+  2.I57 

i38  4i    4.2 

+  16.08 

602 

33o3 

35  Hydrae,               i 

5 

32  11.75 

3.072 

9o  27  52.  9 

16.14 

6o3 

33n 

38  Hydrae,               K 

5 

33     7.26 

2.881 

io3  39  i3.o 

16.07 

6o4 

33i2 

1  4  Leonis,              o 

4 

33     8.53* 

3.228 

79  25  40.7* 

i6.i3 

6o5 

33i5 

28  Ursae  Majoris, 

5 

34  19.08* 

4.725 

25  39  37.8* 

i6.i5 

606 

3320 

Carinae,              m 

5 

35  H.56 

i.65i 

i5o  39     2.7 

i6.25 

607 

333i 

17  Leonis,               e 

3 

37  19.69* 

3.425 

65  32   16.2* 

i6.33 

608 

3346 

29  Ursae  Majoris,    v 

4 

4o  16.98* 

4-353 

3o  i5  33.o* 

i6.63 

609 

3353 

Carinae,              I 

5 

4i     7.5i 

1.637 

i5i  49     4.4 

i6.57 

610 

3358 

3o  Ursae  Majoris,   <j> 

5 

4i  5i.86* 

4.i5o 

35   i4  17.6* 

16.57 

611 

3365 

Argus,               v 

3 

43  21.18 

1.502 

i54  22  38.i 

i6.58 

612 

337i 

24  Leonis,              /* 

3 

44  i3.4i* 

3.428 

63   17  2i.3* 

16.70 

6i3 

3372 

39  Hydrae,              v1 

5 

44  i5.88 

2.885 

io4     8  4i.i 

16.66 

6i4 

34io 

Argils,               <j> 

4 

5i  36.23 

2.092 

i43  5i    i9.o 

16.96 

6i5 

34i5 

29  Leonis,               TT 

4* 

52  17.01* 

3.182 

81   i4  18.4* 

17.06 

616 

3446 

21  Leonis  Minoris, 

5 

58  34.i4* 

3.57i 

54     i  35.i* 

17.31 

617 

3453 

3o  Leonis,              77 

3* 

59     8.85* 

3.285 

72  3o  29.5* 

i7.32 

618 

3457 

3  1   Leonis,              A 

5 

59  56.  27* 

3.194 

79  16     9.5* 

17-42 

619 

3458 

1  5  Sextantis, 

5 

10     o.i5.59 

3.077 

89  38  26.0 

i7.39 

620 

3459 

32  Leonis,              a 

i 

o  22.74* 

3.206 

77  18     6.8* 

17.  39 

621 

3473 

4  1  Hydrae,               /I 

41 

3  i6.75 

2.927 

101  36  5i.8 

17.60 

622 

3495 

Ursae  Minoris, 

5i 

6  58.  81* 

10.207 

4  59  29.4* 

17.73 

623 

3496 

32  Ursae  Majoris, 

5 

7    4.60 

4.468 

24    8  46.2* 

17.71 

624 

35o5 

33  Ursae  Majoris,   A 

ft 

8     1.86* 

3.659 

46  20  20.  o* 

17.77 

625 

35o8 

36  Leonis,              f 

4* 

8  20.43 

3.355 

65  5o  i3.6 

17.70 

626 

35o9 

Velorum,            q 

4 

8  27.11 

2.5i3 

l3l    22    52.7 

17.  91 

627 

35i6 

Argus,               w 

4 

10    10.  II 

1.426 

i59  17  38.o 

17.77 

628 

3523 

4  1   Leonis,               y 

2 

ii  41.78* 

3.322 

69  24     7.0* 

18.01 

629 

3526 

Carinae,               q 

5 

12       4.94 

1.986 

i5o  35     2.2 

17.87 

63o 

3528 

Draconis, 

51 

12  i5.65 

8.i32 

6  4o  57.7* 

17.  95 

63i 

353i 

Ursae  Majoris, 

5 

i3  i5.2o 

4.445 

23  4o  38.4 

17-98 

632 

3533 

34  Ursae  Majoris,   p 

3 

i3  22.36* 

3.6i5 

47  44  53.3* 

17.  89 

633 

3536 

Velorum,           V 

5 

i3  59.05 

2.239 

i  44  16  39.4 

i8.o9 

634 

3546 

Velorum,           T 

5 

l5    2%).  25 

2.  196 

i45  17  27.3 

i8.34 

635 

3552 

Velorum,            r 

5 

i5  54.07 

2.558 

i3o  53  5o.i 

18.08 

636 

356o 

3o  Leonis  Minoris, 

41 

17  i8.33 

3.470 

55  26  3i.i 

18.18 

637 

3568 

42  Hydrae,              /u 

4 

18  5o.4o 

2.900 

i  06    4  i9-4 

18.26 

638 

3572 

3  1  Leonis  Minoris,  (3 

41 

19  ii  .64* 

3.5o2 

52  3i    33.i 

18.26 

639 

3578 

Antliae,               a 

41 

20  17.69 

2.736 

120  18  23.4* 

18.24 

64o 

358o 

36  Ursae  Majoris, 

5 

20    59.82* 

3.911 

33  i5     8.1* 

18.19 

64i 

3585 

Carinse              I 

41 

21     24.76 

I.  212 

63  16     7.0 

18.24 

642 

3586 

Carinae, 

5 

21  4i  .59 

1.885 

54  52  38.i 

18.24 

643 

3589 

Velorum,            P 

5 

21   5o.35 

2.227 

46  52  24.0 

17.92 

644 

3594 

Carinae,              s 

5 

22    22.58 

2.161 

4?  58  25.3 

18.23 

645 

3607 

Ursae  Majoris, 

5 

24    27.94 

3.544 

48  48  i5.3 

i8.34 

646 

3609 

4?  Leonis,               p 

4 

24  54.58* 

3.171 

79  55  24.0* 

i8.39 

647 

36io 

34  Leonis  Minoris, 

5 

24  55.62 

3.459 

54  i4  25.5 

i8.37 

648 

36i2 

37  Ursae  Majoris, 

5 

25    27.85* 

3.939 

32     8  49.3* 

18.39 

64o 

36i9 

Carinae,              p 

4 

26  42.i3 

2.  Ill 

5o  54  52.4 

i8.38 

65o 

364o 

37  Leonis  Minoris, 

41 

o  3o  16.07 

4-3.4o3 

57  i4  46.3 

+  !8.54 

TABLE   XXX. 


425 


CATALOGUE   OP    1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

b' 

c'                d' 

601 

-8.9o58 

+8.785o 

+0.3329 

+8.7815 

—  9.8692 

—9.7773 

+  I.2038 

+9.7808 

602 

8.7266 

8.6029 

0.4863 

+6.6356 

—9.6424 

—7.8117 

.2049 

9.7790 

6o3 

8.74o3 

8.6i3o 

0.4589 

+8.n33 

—9.75o4 

—9.2770 

.2062 

9.7766 

6o4 

8.  7353 

8.6079 

0.5078 

—7.9989 

—  9.5009 

+9.1675 

.2062 

9.7766 

6o5 

9.0930 

8.9610 

0.6742 

—9.0479 

+9.5406 

+9.86o5 

.2079 

9-7736 

606 

9  .  o4o4 

8.9049 

0.2217 

+8.9808 

—9.8618 

—  9.8472 

.2091 

9-7713 

607 

8.7745 

8.63o5 

0.5347 

—8.3916 

—9.1864 

+9.5269 

.2120 

9.7657 

608 

9.o353 

8.8794 

o.64i8 

—8.9717 

+  9.4408 

+  9-85oi 

.2l6o 

9.7578 

609 

9.0646 

8.9052 

0.2173 

+9.0098 

—9.8494 

—9.8601 

.2171 

9.7555 

610 

8.9786 

8.8i63 

0.6174 

—8.8907 

+9.8276 

+9.8279 

.2180 

9.7535 

6n 

9.1057 

8.9374 

0.1776 

+9.0608 

—9.84l4 

—9.8728 

.22OO 

9.7494 

612 

8.7918 

8.6199 

0.5374 

-8.4445 

-9.i358 

+9.57i6 

.2211 

9.7469 

6i3 

8.7562 

8.584i 

0.4598 

+8.1442 

—9.7460 

—  9.3070 

.2211 

9.  7468 

6i4 

8.9811 

8.7786 

o.32i8 

+8.8883 

—9.8358 

—  9.835i 

.23O2 

9.7254 

6i5 

8.7578 

8.5524 

o.5o24 

—7.9406 

—  9.  5420 

+9.iu5 

.2310 

9.7234 

616 

8.85i8 

8.6194 

o.55i6 

—8.6207 

—8.7348 

+9.7049 

.2382 

9.7086 

617 

8.7811 

8.5462 

o.5i62 

—8.2590 

—9.4193 

+9-4i46 

.2388 

9.7017 

618 

8.7690 

8.53o7 

o.5o48 

—  8.o39o 

—  9.  5228 

+9.2075 

.2397 

9.6992 

619 

8.7617 

8.5220 

0.4878 

—6.5592 

—9.  6344 

+7.7353 

.2400 

9.6981 

620 

8.7726 

8.5324 

o.5o8o 

—8.1147 

-9-4967 

+9.2800 

.2402 

9.6977 

621 

8.7740 

8.  5209 

0.4678 

+8.0779 

—9.7188 

—  9.245o 

.2433 

9.6880 

622 

9.8292 

9.5598 

1.0137 

—9.8275 

+9.6375 

+9.9432 

.2471 

9.6754 

628 

9.i57i 

8.8870 

o.65i4 

—9.1174 

+9«4o5o 

+9.9052 

.2472 

9-6749 

624 

8.9105 

8.636o 

0.5646 

—  8.7496 

+8.oo43 

+9.785i 

.2482 

9.6715 

625 

8.8100 

8.534i 

O.5252 

—8.4221 

—  9.3io3 

+9.5584 

.2485 

9.6704 

626 

8.8951 

8.6187 

o.4oi3 

+8.7i53 

—9.8026 

—9.7666 

.2486 

9.67OO 

627 

9.2235 

8.9392 

o.i582 

+9.1945 

—9.7669 

—9.9191 

.25o3 

9.6638 

628 

8.8022 

8.5io8 

o.5i84 

—8.3485 

—9.39ii 

+9.4959 

.25i8 

9.  6582 

629 

9.0827 

8.7894 

0.2999 

+9.0227 

-9.784i 

—  9.8900 

.2522 

9.6567 

63o 

9.7081 

9.4i4o 

0.9158 

—  9.7o5i 

+  9.6014 

+9.9472 

.     .2524 

9.656i 

63i 

9.1712 

8.  8725 

0.6474 

—  9.i33o 

+9.3748 

+9.9129 

.2533 

9.6524 

632 

8.9058 

8.6o65 

0.5582 

—8.  7334 

—  8.2900 

+9.7788 

.2534 

9.65i9 

633 

9.0094 

8.7072 

o.35o4 

+8.9189 

—9.7894 

—  9.86i3 

.254o 

9.6496 

634 

9.  02  i  5 

8.7129 

0.3462 

+8.9364 

—9.7850 

—9.8680 

.2553 

9-6444 

635 

8.8990 

8.  5877 

0.4087 

+8.7i5i 

—9.7891 

—9.7696 

.2558 

9.6422 

636 

8.863i 

8.545o 

o.54o3 

—8.6169 

—9.0477 

+9.7086 

.2671 

9.6367 

637 

8.7975 

8.4719 

0.4633 

+8.2397 

—9.7286 

—9.3985 

.2585 

9.63o6 

638 

8.8809 

8.5535 

0.5449 

—  8.665o 

—  8.93io 

+9.74o8 

.2588 

9.6292 

639 

8.8453 

8.5i25 

0.4379 

+8.5482 

—9.7689 

—9.6605 

.2598 

9.6248 

64o 

9.o43o 

8.7068 

o.5938 

—8.9654 

+9.o5oo 

+9.88o5 

.2604 

9.62i9 

64  1 

9.3232 

8.9849 

0.0849 

+9-3o44 

—9.7196 

—9.9397 

.2607 

9.62O2 

642 

9.i547 

8.8i5o 

0.2752 

+9.1116 

—9.  7488 

—  9.9166 

.2610 

9.6i9o 

643 

9.0452 

8.7047 

0.3463 

+8.9682 

—9.7672 

—9.8819 

.2611 

9.6184 

644 

9.0587 

8.7i55 

o.3399 

+8.98-70 

—9.7637 

-9.8877 

.2616 

9.6162 

645 

8.9085 

8.5547 

0.5495 

—8.7272 

-8.7767 

+9.7798 

.2633 

9.6073 

646 

8.7922 

8.436i 

o.5oo6 

—  8.o35i 

-9.5524 

+9.2044 

.2637 

9.6054 

647 

8.8761 

8.52oo 

o.5389 

—8.6428 

—  9.0671 

+9.7282 

.2637 

9-6o54 

648 

9.0599 

8.7010 

0.5937 

—8.9876 

+9.o354 

+9.8897 

.2642 

9«6o3o 

649 

9.1001 

8.7348 

0.3261 

+9.0416 

-9.7459 

—9.90.44 

.2652 

9.5976 

65o 

—8.8649 

+8.4807 

+o.53i3 

—8.5982 

—  9.2000 

+9.6991 

+    .2681 

+9.58i6 

426 


TABLE  XXX. 


CATALOGUE  OP  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.        s. 

s. 

0              /                ~it 

// 

65i 

3644 

Velorum,           p 

5 

10  3o  59.76 

+2.436 

i37  26  48.3 

+  I8.38 

652 

3646 

Hydrae,               03 

5 

3i   16.67 

2.920 

106     5  56.5 

i8.53 

653 

3647 

38  TJrsae  Majoris, 

5 

3i   89.19 

4.214 

23  3o     o.i* 

18.66 

654 

3652 

Ursae  Majoris, 

5 

32     l5.29* 

4.435 

20       8    28.  7* 

i8.65 

655 

3655 

Carinae,              p 

5 

33     2.82 

2.244 

i48  24  12.3 

18.67 

656 

366o 

Chamaeleontis,  y 

5 

33  39.o4 

0.732 

16-7  49  48.3 

i8.56 

657 

3685 

£2  Leonis  Minoris, 

41 

37  3o.86 

3.36i 

58  3i  42.0 

18.79 

658 

3686 

Argus,                0 

3 

37  37.i9 

2.118 

i53  36  33.5 

18.76 

659 

3695 

Argus,                 rj 

2 

39  i5.44 

2.3o5 

i48  53  49.5 

i8.73 

660 

3702 

Argus,                ft 

3 

4o  19.80 

2.56o 

i38  37  41.2 

18.92 

661 

37i5 

Hydrae,               v 

4 

42  i3.55 

2.954 

io5  24  37.i 

18.74 

662 

3724 

Chamaeleontis,  d* 

5 

44  19.32 

o.63i 

169  44  57.9 

18.97 

663 

3728 

46  Leonis  Minoris, 

41 

44  54.49 

3.379 

54  58  39.8* 

i9.25 

664 

3729 

45  Ursae  Majoris,    w 

5 

45  19.44* 

3.491 

46     o  46.4* 

19.0!: 

665 

374o 

Carinae,              u 

5 

47  24.97 

2.396 

i48     3  26.9 

19.09 

666 

374a 

54  Leonis, 

41 

4y  29.24 

3.275 

64  27    4-4 

I9'°7 

667 

3766 

7  Crateris,             a 

4 

62  28.25 

2.919 

io7  3o     1.8 

19.05 

668 

3767 

48  Ursae  Majoris,   (3 

2 

52  45.  4i* 

3.685 

32  48  53.6* 

19.  17 

669 

3768 

58  Leonis,               d 

5 

62  48.71* 

3.io4 

85  34  42.3* 

19.25 

670 

3776 

60  Leonis,               b 

5 

54  19.05 

3.219 

69     o  56.  9 

19.20 

671 

3777 

5o  Ursae  Majoris,    a 

It 

54  25.72* 

3.784 

27  26  26.0* 

19.33 

672 

3788 

63  Leonis,               % 

41 

57  16.61* 

3.io3 

81  5i   i5.6* 

19.39 

673 

3793 

Hydrae,               Xl 

5 

58     6.98 

2.884 

116  29     5.i 

19.36 

674 

3794 

Hydra?,               *> 

5 

58  41.71 

2.900 

116  28  4o.7 

19.  37 

675 

38i2 

52  Ursae  Majoris,   + 

31 

ii     i   12.67* 

3.4n 

44  4i  i9.6* 

19.49 

676 

677 

38i5 

3822 

Hydra, 
Hydrae, 

5 
5 

i  29.16 
2  4o.85 

2.89-7 
2.856 

ii7  16     5.4 
121   33   i6.3 

19.43 
19.55 

678 

3826 

ii   Crateris,            /? 

4 

4  17-27 

2.947 

112       O    26.4 

i9.57 

679 

3834 

68  Leonis,               6 

2l 

6     7.46* 

3.209 

68  39  i9.3* 

i9.6^ 

680 

3838 

70  Leonis,               6 

3 

6  21.99 

3.i6i 

73  45     o.7 

I9.52 

68  1 

3842 

72  Leonis, 

5 

7  i3.i6 

3.207 

66     5  i5.4 

i9.5^ 

682 

3848 

74  Leonis,               <j> 

5 

9       2.12* 

3.o53 

92  49  57.3* 

i9.6i 

683 

385i 

53  Ursae  Majoris,    £ 

4 

IO    IO.46 

3.223 

57  37  36.4 

20.  l5 

684 

3852 

54  Ursae  Majoris,    v 

4 

IO    22.  o5 

3.268 

56     5  16.2 

19.55 

685 

3856 

55  Ursae  Majoris, 

5 

10  56.  5i* 

3.3o2 

5o  59  82.2* 

19.68 

686 

3859 

12  Crateris,            6 

31 

ii  5o.65* 

2-997 

io3  58     3.4* 

19.44 

687 

3862 

77  Leonis,               a 

4 

i3  23.98* 

3.099 

83     8  58.i* 

19.66 

688 

3866 

Centauri,            IT 

4 

i4  10.55 

2.758 

i43  4o  i4.5 

19.98 

689 

387? 

78  Leonis,              t 

4 

i  6     6.12* 

3.i37 

78  38  4i.5* 

19.  75 

690 

388i 

1  4  Crateris,             e 

5 

17     2.28 

3.028 

IOO       2     I  5.  4 

19.69 

691 

3883 

1  5  Crateris,            y 

4 

17  23.56 

2.990 

106  5i   36.i 

19.67 

692 

3885 

Ursae  Majoris, 

5 

17  27.47* 

3.439 

33   19  4i.o* 

19.  67 

693 

Sgoo 

84  Leonis,               r 

4 

20   i3.3i» 

3.091 

86   19     5.9* 

19-78 

694 

39i4 

i   Draconis,           /I 

31 

22    26.42* 

3.67i 

19  5o  3o.4* 

19-87 

695 

3916 

87  Leonis,               e 

41 

22    39.03* 

3.o66 

92   10  35.9* 

19.84 

696 
697 

3922 
3928 

Hydrse, 
Hydrae, 

5 

4 

24    So.gi 

25  38.25 

2.964 
2.941 

118  26  22.1* 
121     i  39.2 

19.  67 
19.86 

698 

394i 

Centauri,            "k 

4 

28  53.66 

2.724 

i52  ii  24.0 

19.87 

699 

3943 

21  Crateris,             0 

4 

29     4.66 

3.043 

98  58  2i.5 

19.86 

700 

3946 

9i   Leonis,                v 

41 

ii  29  16.07* 

4-3.  o74 

89  59  46.6* 

+  19.87 

TABLE   XXX. 


427 


CATALOGUE    OF   1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b         \         c 

d 

a' 

b' 

c' 

# 

65i 

—8.9602 

+8.6721 

+o.4oi3 

+8.8274 

—9.7677 

—9.8337 

+  1.2686 

+9.5783 

652 

8.8079 

8.4182 

o.466i 

+8.2608 

—9.7173 

—9.4096 

1.2688 

9.577o 

653 

9.1901 

8.7984 

0.6260 

—  9.  i525 

+9.2360 

+9.9293 

I  .269I 

9.6762 

654 

9.2543 

8.8593 

0.6466 

—  9.2269 

+9.3o38 

+9.9400 

i.2696 

9.6724 

655 

9.0726 

8.6733 

0.3552 

+9.0029 

-9.7347 

-9.8983 

i  .2702 

9.6686 

656 

9.4684 

9.o658 

9.8976 

+9.4686 

—9.  6544 

—9.9686 

1.2706 

9.  5658 

657 

8.8643 

8.4398 

0.5262 

—8.6820 

—  9.27oo 

+9.6890 

1.2735 

9.5468 

658 

9  -i474 

8.7223 

0.3269 

+9.0996 

—  9.7o59 

-9.9235 

1.2735 

9.  5463 

659 

9.0832 

8.6487 

0.3628 

+9.0168 

—  9.7i58 

—  9.9061 

1.2747 

9.5379 

660 

8.9770 

8.536i 

o.4o7i 

+8.8523 

-9.7343 

—9.  8485 

1.2764 

9.5323 

661 

8.8i43 

8.8621 

0.4695 

+8.2388 

—  9.7o49 

—9.3989 

1.2767 

9.5223 

662 

9.5495 

9.o845 

9.8283 

+9.5426 

—  9.598o 

—  9.9689 

1.2781 

9.5io9 

663 

8  .  886c 

8.4i83 

o.5279 

—  8.6457 

—  9.2322 

+9-7'35i 

1.2786 

9.5o76 

664 

8.  943^ 

8.4722 

o.542o 

—8.  7861 

—8.9638 

+  9.8182 

1.2788 

9.5o53 

665 

9.0783 

8.5938 

o.38o7 

+9.0069 

—  9.6936 

—  9.9066 

i  .2801 

9.4934 

666 

8.8465 

8.36i6 

o.5i45 

—  8.48i3 

—  9.410^ 

+9.6127 

1.2801 

9.4930 

667 

8.8254 

8.3o75 

0.4696 

+8.3o35 

—  9.7ooo 

—9.4690 

i.283i 

9.463o 

668 

9.0710 

8.55i2 

o.5647|—  8.9955 

+7.8976 

+9.9066 

1.2833 

9.4612 

669 

8.8o63 

8.2861 

0.4914—7-6933 

—  9.6i32 

+8.8681 

1.2833 

9.46o9 

670 

8.8356 

8.3o5o 

o.5o73 

—  8.3896 

—9.4829 

+9.5369 

1.2841 

945i3 

671 

9.1424 

8.6110 

o.5795 

—  9.0906 

+8.7033 

+9.9301 

1.2842 

9  .4606 

672 

8.8118 

8.2601 

0.4945 

—7.9632 

—9.6926 

+9-i349 

1.2868 

9.43:8 

673 

8.856o 

8.2982 

o.46i3 

+8.5o53 

—9.706^ 

—  9.6333 

1.2862 

9.4261 

674 

8.8563 

8.2942 

o.46i6 

+8.5o55 

—9.7043 

—9-6335 

1.2866 

9.4221 

675 

8.9624 

8.38n 

0.5332 

—  8.8142 

—  9.0962 

+9-8374 

1.2878 

9.4o43 

676 

8.8608 

8.2774 

0.4619 

+8.6218 

—  9.7002 

-9.6467 

1.2879 

9.4o24 

677 

8.8797 

8.287o 

o.4575 

+8.5984 

—9.6984 

—  9.7o5o 

1.2886 

9.3936 

678 

8.8438 

8.2383 

0.4684 

+8.4176 

—  9.692^ 

—9.6608 

i.2893 

9.38i5 

679 

8.8427 

8.222O 

o.5o4i 

—  8.4o37 

—  9.5o43 

+9.5490 

I  .29OI 

9.3672 

680 

8.8296 

8.2069 

0.4998 

—8.  2766 

—9.6462 

+9.4349 

I  .2902 

9.  3653 

681 

8.85i3 

8.2213 

o.5o6o 

—  8.459i 

—9.483s 

+9.6962 

I.29o6 

9.3585 

682 

8.8i36 

8.i679 

0.4852 

+7.6076 

—9.6477 

—8.683o 

1.2914 

9.3435 

683 

8.8869 

8.23n 

o.5i23 

—8.  6166 

—9.3997 

+9.7184 

I.29I9 

9.3338 

684 

8.8946 

8.237i 

o.5i37 

—8.64i2 

—  9.38o6 

+9.7363 

I  .292O 

9.3322 

685 

8.9234 

8.2606 

0.6186 

—8.7224 

—9.3o86 

+9.7889 

I  .2922 

9.3272 

686 

8.8273 

8.1662 

0.4773 

+  8.2100 

—9.6734 

-9.373o 

I  .2926 

9.3i93 

687 

8.  8180 

8.1322 

0.4918 

—  7-8946 

—  9.609^ 

+9.o676 

I  .2932 

9.3o52 

688 

9.0425 

8.3493 

o.4329 

+8.9487 

—  9.6202 

—  9.8974 

1.2935 

9.298o 

689 

8.8245 

8.II2I 

0.4944 

—  8.n87 

—  9.5888 

+  9.2862 

1.2942 

9.2796 

690 

8.8229 

8.1010 

0.4809 

+8.0642 

—  9.6626 

—  9.2336 

I  .2945 

9.27o4 

691 

8.8354 

8.1098 

0.4765 

+8.2979 

—9.6711 

—  9.4549 

1.2947 

9.2668 

692 

9.0765 

8.35oi 

o.5373 

—8.9986 

—8.9263 

+  9.9i45 

1.2947 

9.2661 

693 

8.8182 

8.0621 

0.4893 

—7.  6269 

—9.6263 

+  8.8011 

I  .2967 

9.2372 

694 

9.2873 

8.5o57 

0.5653 

—  9.26o7 

+  7.7924 

+9.9676 

I  .2964 

9.2ia6 

696 

8.8i84 

8.o344 

o.486i 

+7.3979 

-9.6434 

-8.5737 

1.2964 

9.2IO2 

696 

8.8746 

8.o638 

o.47i3 

+8.5525 

—9.6674 

—  9.6727 

I.297I 

9.1841 

697 

8.8861 

8.o652 

0.4699 

+8.5983 

—  9  .6622 

—  9.7o73 

1.2973 

9.1743 

698 

9.  i5io 

8.2863 

0.4358 

+9-°977 

—9.6081 

—9.9427 

I  .2982 

9.i3i3 

699 

8.8253 

7.9580 

0.4832 

+8.oi83 

-9.6629 

—  9.  i89i 

1.2983 

9.i288 

700 

—8.8200 

+7.9500 

-f-o.4872 

-4.65i7 

—9.6376 

+  6.8278 

+  1.2983 

+9  .  1261 

428 


TABLE  XXX, 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.       s. 

s. 

~0              '            ~> 

" 

701 

3978 

27  Crateris,             £ 

4 

II    37    IO.o4 

+3.o33 

107  3o  58.5 

+  I9-97 

702 

3979 

2  Virginis,            £ 

5 

37  33.o3 

3.097 

80  54  3o.i 

20.00 

703 

398i 

63  Ursae  Majoris,   % 

4 

38     6.53* 

3.209 

4l     23    20.5* 

19.94 

704 

3982 

3  Virginis,             v     j  41 

38     8.97* 

3.093 

82  37  49.4* 

20.  l6 

7o5 

3984 

Muscae, 

ft 

38  34.44 

2.792 

i55  53  52.! 

20.  03 

706 

399o 

9  3  Leonis, 

4 

4o  i4.66 

3.107 

68  56  49.3 

19.  98 

707 

3995 

94  Leonis,               (3 

2* 

4i  24.27* 

3.o66 

74  35  22.6* 

20.  o9 

708 

4OO2 

5  Virginis,            ft 

31 

42  52.  9o* 

3.128 

87    23    25.2* 

20.28 

709 

4oi5 

Hydrae, 

4 

45  20.47* 

3.oi4 

123       4    26.5* 

20.  o4 

710 

4017 

64  Ursae  Majoris,    y 

2 

45  55.o6* 

3.  202 

35  28  16.8* 

20.  o4 

711 

4o48 

Chamaeleontis,  s 

5 

52  i4.93 

2.83o 

167  23  i3.o 

2O.  IO 

712 

4o52 

8  Virginis,             TT 

5 

53  ii.  16* 

3.079 

82  32  57.7* 

20.  IO 

7i3 

4072 

9  Virginis,             o 

41 

57  34.07* 

3.o64 

80  26     1.4* 

20.  o5 

7i4 

4078 

Crucis,               T? 

41 

59     5.65 

3.o4i 

i53  46  38.6 

20.24 

7i5 

4o87 

Centauri,            6 

3 

12     o  36.54 

3.070 

i39  53  14.1 

20.  l5 

716 

4090 

i   Corvi,                 a 

41 

o  41.27 

3.082 

n3  53  28.0 

20.  IO 

717 

4o97 

2  Corvi,                  e 

4 

2    25.25 

3.077 

in  47     4.5 

20.  o3 

718 

4io3 

Centauri,            p 

4 

3  5o.i5 

3.090 

i4i  3i  57.2 

20.  o3 

719 

4lI2 

Draconis, 

5 

5     6.43* 

2.936 

ii   33     o.3* 

20.  o3 

720 

4l20 

Crucis,               6 

3 

7  12.68 

3.128 

i4y  54  49.3 

20.00 

721 

4i23 

69  Ursae  Majoris,   <5 

3 

7  58.70* 

3.oi6 

32     8     1.7* 

20.  IO 

722 

4124 

4  Corvi,                 7 

3 

8     5.97 

3.077 

106  42  3o.4 

2O.02 

?33 

4i25 

6  Comae, 

5 

8  22.93 

3.o54 

74  i5  53.2 

20.  o5 

724 

4126 

2  Canum  Venat., 

5 

8  35.84* 

3.o34 

48  3o  i5.9* 

2O.O7 

725 

4127 

7  Comae, 

5 

8  44.96 

3.o5o 

65   i3   14.1 

2O.O7 

726 

4128 

Canum  Venat., 

5 

8  57.48 

3.o4i 

56     6     2.8 

2O.22 

727 

4i3i 

Chamaeleontis,  j3 

5 

9  40.27 

3.3o4 

168  28  46.3 

20.  o5 

728 

4i33 

Crucis,               C 

5 

10  20.59 

3.i53 

i53  10  20.8 

20.46 

729 

4i45 

1  5  Virginis,             tj 

31 

12  i3.89* 

3  .  067 

89  49  58.5* 

20.07 

780 

4i5i 

1  6  Virginis,             c 

5 

12  43.  93* 

3.o49 

85  5i     6.5* 

20.09 

73i 

4i56 

1  1   Comae, 

5 

i3     8.i3 

3.o36 

71  22  39.8 

19.  95 

732 

4i58 

Crucis,                e 

4 

i3  18.27 

3.177 

:49  34  i9.6 

i9.88 

733 

4169 

12  Comae, 

5 

i4  57.61 

3.029 

63   i9  i3.3 

20.00 

734 

4i8i 

1  3  Comae, 

5 

16  46.83 

3.  022 

63     4     5.6 

20.02 

735 

4i86 

Crucis, 

41 

18   i3.i8 

3.283 

i52  17  23.6 

20.  01 

736 

4i87 

Crucis,               a 

i 

18   18.00 

3.25i 

i52  i5  69.7 

i9.94 

737 

4191 

1  4  Comae, 

5 

18  53.77 

3.oi5 

61  53  59.6 

20.01 

738 

4i95 

1  5  Comae,                y 

41 

19  27.45 

3.oo4 

60  53  49.1 

20.  O9 

739 

4196 

1  6  Comae, 

5 

19  29.08 

3.oi8 

62  20  33.2 

I9.98 

740 

4197 

Centauri,            a       41 

19  57.03 

3.i85 

i39  23  55.8 

I9.9I 

i 

74i 

4202 

Centauri, 

5 

20    24.87 

3.i55 

128    12    37.2 

20.23 

?42 

4211 

7  Corvi,                (5 

3 

22       6.8l 

3.io6 

io5  4o  46.5 

20.  12 

743 

4216 

Crucis,               7 

2 

22    52.84 

3.275 

i46  16  17.7 

20.  l3 

744 

4224 

Muscae,               7 

4 

23  35.  o5 

3.46o 

161   18   i3.2 

i9.98 

745 

4226 

8  Corvi,                7i 

41 

•24  20.93 

3.o83 

io5  21  52.4 

20.  OI 

746 

4234 

9  Corvi,                 j3 

2l 

26  3o.94* 

3.i3! 

112  34    o.i* 

I9.99 

747 

4235 

8  Canum  Venat,  (3 

4 

26  36.  4o* 

2.865 

47  49  35.5* 

i9.64 

748 

4239 

5  Draconis,           « 

31 

27     3.io* 

2.610 

i9  23    4-4* 

i9.96 

749 

4240 

23  Comae, 

41 

27  22.75 

3.oi5 

66  32  37.3 

i9.9i 

760 

4245 

Muscae,               a 

4 

12    28    l8.20 

+  3.48i 

i58  18  28.7 

4-19.90 

TABLE   XXX. 


429 


CATALOGUE   OF    1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a. 

I 

c 

d 

a' 

V 

c' 

d' 

701 

—8.8424 

+  7.8422 

+0.4812 

+8.3209 

—9.6471 

—  9.4764 

+  i  .3ooi 

+8.9977 

702 

8.8273 

7-8l97 

0.4902 

—  8.0260 

9,6160 

+9.i966 

i  .3ooi 

8.99o3 

7o3 

9.0016 

7.9829 

o.5o72 

—8.8768 

9.3333 

+9.8732 

I  .3OO2 

8.9793 

704 

8.8255 

7.8o6i 

0.4896 

-7.9337 

9.6214 

+9.  1062 

1.3002 

8.9786 

7o5 

9.2110 

8.i83o 

0.4459 

+9.1713 

9.4190 

—  9.9585 

i.3oo3 

8.97oi 

706 

8.8523 

7.7889 

0.4935 

—8.4077 

9.5730 

+9.5538 

i.3oo6 

8.935o 

707 

8.8384 

7-7485 

0.4914 

—8.2628 

9.5987 

+9.423o 

i.3oo8 

8.9o87 

708 

8.8232 

7.6972 

o.4879 

-7.4816 

0.6336 

+8.6572 

i  .3oio 

8.8-728 

709 

8.8998 

7.7061 

o.4793 

+8.6368 

9.6006 

—  9.736i 

i.3oi3 

8.8o55 

710 

9.0694 

7.8484 

o.5o32 

—8.9703 

9  .  3o49 

+9.9100 

i.3oi4 

8.7882 

711 

9-4845 

8.oi38 

o.4577 

+9.4739 

9.  1066 

—9.9891 

I  ,3O2O 

8.6291 

712 

8.8274 

7.3oo6 

0.4880 

—  7.94o3 

9.6298 

+9.112-7 

1.3020 

8.473o 

718 

8.83oo 

6.8554 

o.4876 

—  8.o5o6 

9.6296 

+9.2206 

1.3022 

8.0264 

7i4 

9.1786 

4-6.7754 

o.4857 

+9.i3i5 

9.2906 

—9.9628 

I  .3O22 

+  7.5967 

716 

9.0148 

—6.4392 

o.4878 

+8.8984 

9.4434 

—9.  8835 

I  .3O22 

—  7.4244 

716 

8.8628 

6.34oi 

o.4875 

+8.4703 

9.6973 

—9.6076 

I  .3O22 

7.4773 

717 

8.856i 

6.8799 

o.488o 

+8.4256 

9.6011 

—9.6696 

I  .3O22 

8.0238 

718 

9.o3oo 

7.2537 

0.4912 

+8.9238 

9.4098 

-9.8937 

I  .3O22 

8.2236 

719 

9.5223 

7.8702 

o.466  i 

—  9.5i34 

9.0362 

+9.9910 

I.302I 

8.3478 

720 

9.0984 

7.5964 

0.4966 

+9.0264 

9.3092 

—9.9278 

1.3020 

8.4978 

721 

9.0978 

7.6398 

o.4766 

—  9.0266 

9.4i55 

+9.9276 

I  .3O2O 

8.5417 

722 

8.8424 

7.3909 

0.4892 

+8.3oio 

9.6079 

—  9-4584 

I  .3O2O 

8.5482 

728 

8  .  8402 

7.4o36 

0.4853 

—  8.2735 

9.63io 

+9.433o 

i  .3oi9 

8.563i 

724 

8.9491 

7.  5236 

0.4809 

—8.7704 

9.5439 

+9.8209 

i.3oi9 

8.6742 

725 

8.8655 

7.4475 

0.4839 

—8.4879 

9.6i28 

+9.6220 

i.3oi9 

8.6817 

726 

8.9045 

7.4967 

0.4822 

—  8.6609 

9.582i 

+9.7461 

i.3oi9 

8.59i9 

727 

9.523i 

8.1482 

0.624-7 

+9.5i42 

8.6671 

—9.9908 

i.3oi8 

8.6247 

728 

9.  1690 

7.8238 

o.5o38 

+9.1195 

9.I920 

—  9.9601 

i.3oi8 

8.6543 

729 

8.8233 

7.55io 

o.4872 

—6.2886 

9.6377 

+7-4646 

i.3oi6 

8.7271 

73o 

8.8244 

7.5696 

0.4865 

—7.  6838 

9.64o3 

+8.8687 

i.3oi6 

8.7445 

73i 

8.8466 

7.6o53 

0.4836 

—  8.35o8 

9.633o 

+9,5o35 

i.3oi5 

8.  7680 

732 

9.1186 

7.8829 

o.5o55 

+9.0543 

9.2299 

—9.9349 

i.3oi5 

8.7636 

733 

8.8719 

7.6872 

o.48io 

—8.524i 

9.62O2 

+9.65i3 

i.3oi3 

8.8i44 

734 

8.8726 

7.738o 

0.4801 

-8.5286 

9.6233 

+9.6649 

i.3on 

8.8642 

735 

9.i55i 

8.o564 

o.5i49 

+9.1022 

9.n89 

—9.9457 

i  .3oo9 

8.8999 

736 

9.i54y 

8.o578 

o.5i5o 

+9.1018 

9.n83 

-9.9456 

i.3oo8 

8.9oi6 

737 

8.8769 

7-794i 

o.4788 

—8.5499 

9.6249 

+9.67i6 

i.3oo7 

8.9i57 

738 

8.8810 

•7.8109 

0.4782 

—  8.5679 

9.6238 

+9.6854 

i  .  3oo7 

8.9284 

739 

8.875o 

7.8o56 

o.4787 

—  8.54i7 

9.  62-72 

+9.665i 

i.3oo7 

8.929o 

740 

9.0088 

7.9497 

o.5o6o 

+8.8892 

9.3353 

-9.8787 

i  .  3oo6 

8.9392 

74  1 

8.9269 

7.8778 

o.5oo? 

+8.7i83 

9.4564 

-9.7896 

i.3oo5 

8.9492 

742 

8.838; 

7.8242 

0.4923 

+  8.27OI 

9.593i 

—  9.4298 

1.3002 

8.9838 

743 

9.0773 

8.0780 

o.5i46 

+8.99-72 

9.  1989 

—9.9178 

i.Sooi 

8.9986 

744 

9.3i57 

8.3297 

o.54ii 

+9.2922 

8.6212 

—9.9742 

1.2999 

9.0117 

745 

8.8373 

7.8652 

0.4927 

+8.2604 

9.59i4 

—9.4207 

i.2998 

9.0266 

746 

8.8556 

•7.9209 

0.4962 

+8.4396 

9.55i9 

—9.6811 

1.2993 

9.o624 

747 

8.9511 

8.0179 

0.4669 

-8.7781 

9.6oiZ 

+9.8240 

1.2993 

9.o639 

748 

9.2999 

8.3739 

o.4i88 

—9.2745 

9.4o67 

+9.97i6 

1.2992 

9.o7io 

749 

8.8583 

7.9376 

o.4773 

—8.4582 

9.6489 

+9.6968 

1.2991 

9.0762 

75o 

—  9.2628 

—8.3467 

+0.5422 

+9.2209 

—8.  6618 

—  9.9648 

+1.2989 

—  9.  o9o6 

430 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.   m.         s. 

s. 

0                               // 

" 

75i 

425l 

Centauri,           r 

5 

12  29  31.49 

+3.24i 

i37  42  49.2 

+  19.84 

762 

4257 

26  Virginis,             % 

5 

3i   3o.74 

3.094 

97  I0     7-4 

'    I9-9I 

753 

4262 

Centauri, 

5 

3i  46.  59 

3.220 

i29     9  37.5 

I9.92 

754 

4264 

Centauri,            7 

3 

33  i6.3i 

3.266 

i  38     8     6.9 

19.89 

755 

4268 

29  Virginis,            7 

4 

34    3.69* 

3.o4o 

90  37  33.  7* 

19.  85 

756 

4271 

3o  Virginis,            p 

5 

34  17.41 

3.o37 

78  56     9.4 

19.92 

757 

4280 

Muscae,              /? 

4 

37     8.73 

3.577 

i57  i7     8.2 

19.80 

758 

4289 

Crucis,               /? 

2 

38  59.94 

3.443 

i48  52     0.4 

19.71 

759 

4290 

27  Comae, 

5 

39     9.o7 

3.oo2 

72  36     4.7 

19.75 

760 

4293 

Octantis,            t 

5 

39  45.  57 

5.396 

i74  18  14.1 

19.42 

761 

4321 

Centauri, 

5 

45     8.75 

3.287 

129  21  42.1 

19.77 

762 

4325 

Centauri, 

5 

45  49.29 

3.473 

i46  21  4o.3 

19.^7 

763 

4328 

35  Comes, 

5 

45  54.4i* 

2.960 

67  56   19.3* 

19.69 

764 

433o 

4o  Virginis,             ^ 

5 

46  33.4i* 

3.ii6 

98  43  23.4* 

19.66 

765 

4335 

77  Ursae  Majoris,    e 

3 

47  24.86* 

2.668 

33  i3  3o.6* 

19.69 

766 

4339 

Ursae  Minoris, 

51 

47  57.88* 

o.3o4 

5  45  58.  9* 

19.63 

767 

434o 

43  Virginis,             6 

3 

48    2.89* 

3.023 

85  47   ii.  6* 

19.  7i 

768 

4342 

Ursae  Minoris, 

51 

48     5.49* 

0.288 

5  46  17.0* 

19.60 

769 

4346 

12  CanumVenat.,  a 

a* 

49    0.17* 

2.8a3 

5o  52  i3.9* 

19.  56 

770 

435i 

36  Comae, 

41 

5i  3o.25 

2.974 

71  46  48.  7 

19.49 

771 

4353 

Muscae,               6 

4 

52     2.64 

3.995 

160  44  16.9 

19.55 

772 

436o 

37  Comae, 

5 

53     5.57 

2.885 

58  24  i3.4 

19.50 

773 

4366 

78  Ursae  Majoris, 

5 

54  16.67* 

2.601 

32  49  26.6* 

19.48 

774 

4367 

47  Virginis,             e 

3 

54  42.83 

2.993 

78  i3  58.3 

i9.46 

775 

4379 

Centauri,            £2 

5 

58  ii.  o3 

3.464 

i39     6     5.4 

19.62 

776 

4384 

1  4  CanumVenat., 

5 

58  43.20* 

2.825 

53  23  5o.6* 

19.39 

777 

4387 

39  Comae, 

5 

59     2.53 

2.932 

68       2    22.6 

19.42 

778 

4390 

4  1    Comae, 

4 

59  58.  69* 

2.888 

61   34     8.4* 

i9.46 

779 

439r 

49  Virginis,            g 

5 

i  3     o     2.60* 

3.!37 

99  56  13.7* 

19.40 

780 

4395 

45  Hydrae,               ^ 

41 

o  Sg.Si 

3.2I9 

112  18  5o.9 

19.41 

781 

44oi 

5  1  Virginis,            0 

41 

2    11.26* 

3.101 

94  44  12.9* 

19.37 

782 

44o6 

42  Comae,                a 

41 

2    4l.49 

2.0.24 

71  4o  33.6* 

19.1-7 

783 

4409 

Centauri, 

5 

2    5o.O2 

3.398 

i32  34     4.i 

19.38 

784 

44i8 

53  Virginis, 

5 

4    4.93* 

3.i8i 

io5  23   17.1* 

19-57 

785 

4421 

43  Comae,                j3 

41 

4    52.01* 

2.811 

61   21   37,3* 

i8.35 

786 

4426 

Muscae,               TJ 

5 

5     9.35 

3..943 

167     5  45.4 

19.09 

787 

4433 

CanumVenat., 

5 

6  54.62* 

2.734 

49     3     6.7* 

19.  27 

788 

4449 

6  1   Virginis, 

41 

10  33.88* 

3.129 

107  28  3i.6* 

2O.  I^j 

789 

445o 

46  Hydrae,               7 

4 

10  46.68 

3.247 

112    22    39.5 

19.11 

79° 

445  1 

20  CanumVenat., 

5 

10  48.56* 

2,.7o3 

48  38  io.5* 

19.08 

791 

4456 

2  1   Canum  Venat.  , 

5 

ii  5i.25 

2.57I 

39  3i  4i.2 

19.11 

792 

4458 

Centauri,            t 

3 

12    11.09 

3.348 

125     55     12.2* 

19.21 

793 

448o 

6j  Virginis,             a 

i 

17    17.79* 

3.i52 

ioo  22  36.5* 

l8.98 

794 

4483 

Octantis,             K 

5 

17    38.76 

8.o75 

175     o  59.7 

I9.54 

795 

4484 

79  Ursae  Majoris,    f 

3 

17  52.63* 

2.437 

34  17  24.0* 

18.95 

796 

4492 

68  Virginis,            i 

5 

18  48.23 

3.i6i 

101  55  3o.5 

18.92 

797 

4493 

80  Ursae  Majoris,   g 

5 

19  12.48* 

2.4s4 

34  i3  44.o* 

18.92 

798 

45o7 

Centauri, 

41 

22    21.97 

3.445 

128  37  48.8 

18.82 

799 

4532 

"79  Virginis,             C 

4 

27     3.24* 

3.o55 

89  49  38.2* 

18.57 

800 

4538 

24  CanumVenat., 

5 

i3  28  18.98* 

+2.466 

4o  12  56.5* 

+  l8.6o 

TABLE   XXX. 


431 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b        |         c 

d 

a.' 

b' 

c' 

d! 

7fi 

—  8  .992^ 

—8.io48 

+o.5i32 

+8.8616 

—9.2946 

—9.8666 

+  1.2986 

—9.1088 

702 

8.8232 

•7.9643 

0.4906 

+7-9194 

—9.6166 

—  9.0921 

1.2981 

9.i369 

753 

8.9302 

8.o749 

0.6080 

+8.73o6 

-9.3966 

—9.7962 

I  .2980 

9.1406 

754 

8.9960 

8.1699 

o.5i67 

+8.8669 

—  9.2698 

—9.8674 

1.29-76 

9.  1604 

755 

8.8191 

7.9944 

0.4876 

+0.8677 

—9.6358 

—8.0337 

1.2974 

9.1706 

756 

8.8272 

8.0064 

0.4817 

—  8.no3 

—  9.6575 

+9.2782 

i.2973 

9.1734 

757 

9.2314 

8.445o 

0.5546 

+9.1964 

—  8.27i8 

—9.9692 

1.2966 

9.2078 

758 

9.  io4i 

8.3392 

o.5372 

+9.o365 

—8.8998 

—  9.9261 

1.2969 

9.2288 

759 

8.8379 

8.0747 

0.4770 

—  8.3i36 

—  9.6674 

+9-4693 

I  .2969 

9.23o5 

76o 

9.8206 

9.0643 

o.73n 

+9.8186 

+9.1096 

—9.9913 

1.296-7 

9.237i 

76i 

8.92-72 

8.2273 

0.6166 

+8.7294 

—9.3261 

-9.7938 

i.2937 

9.2916 

762 

9.o7i7 

8.3784 

o.54o3 

+8.9921 

—  8.8756 

-9.9117 

1.2935 

9.2980 

763 

8.8482 

8.i557 

0.4717 

—8.4229 

—  9.6784 

+9.6660 

1.2935 

9.2988 

764 

8.8199 

8.i338 

o.493o 

+8.0008 

—  9.6002 

—9.1718 

I  .2932 

9-3o48 

765 

9.o758 

8.3978 

0.4235 

—8.9983 

—9.6124 

+9.9181 

I  .2929 

9.3126 

766 

9.8123 

9.i393 

9.6061 

—  9.8101 

—9.3986 

+9.9882 

I  .2926 

9.3i75 

767 

8.8i55 

8.i434 

0.4843 

—7.6816 

—9.6614 

+8.8565 

I  .2926 

9.3i83 

768 

9.8119 

9.  i4oi 

9.4994 

—  9.8097 

-9.3993 

+9.9882 

1.2926 

9.3i86 

769 

8.9242 

8.2610 

0.4533 

—8.7243 

—  9.6726 

+9.7901 

I  .2922 

9.3268 

77° 

8.8352 

8.i943 

o.473i 

—  8.33o3 

—  9.683o 

+9-484o 

I  .2912 

9.348o 

771 

9.2942 

8.6680 

0.6946 

+9.2692 

+8.84o7 

-9.o637 

I  .2909 

9.3626 

772 

8.  8818 

8.2646 

0.469-7 

—  8.6011 

—  9.69oi 

+9.7076 

1.2906 

9.36io 

773 

9.o776 

8.46o3 

o.4i23 

—  9.0020 

—  9.6359 

+9.9122 

1.2899 

9.37o4 

774 

8.8206 

8.2069 

o.4778 

—  8.i3oi 

—  9-6749 

+9.2970 

1.2897 

9.3738 

776 

8.9937 

8.4o78 

0.5389 

+8.8722 

—  8.9745 

—9.8643 

1.2881 

9.4000 

776 

8.9049 

8.3232 

0.4601 

—  8.68o3 

—9.6986 

+9.7610 

1.2878 

9.4039 

777 

8.8421 

8.2628 

0.4673 

—  8.4i49 

—  9-697i 

+9.5583 

1.2877 

9.4062 

778 

8.8647 

8.2926 

0.4699 

—8.5424 

—  9.7o3i 

+9.6627 

1.2872 

9.4129 

779 

8.8i54 

8.2438 

0.4967 

+8.0624 

—  9.583o 

—9.2219 

I  .2871 

9.4i33 

780 

8.8422 

8.2776 

o.5o72 

+8.4216 

—9.4794 

—9.6639 

1.2867 

9.4199 

781 

8.8092 

8.2536 

0.4914 

+7.7261 

—9.6132 

—  8  .9007 

1.2860 

9.4282 

782 

8.83oi 

8.278i 

o.47oo 

—8.3276 

—9.6963 

+  9.4810 

1.2868 

9.43i6 

783 

8.9402 

8.3893 

o.53i9 

+8.7706 

—  9.i35i 

-9.8i37 

1.2867 

9.4326 

784 

8.8226 

8.28o7 

o.5oi4 

+8.2464 

—9.538i 

—9.4066 

1.2860 

9.4409 

785 

8.863o 

8.3266 

0.4574 

—8.5436 

—  9.7116 

+  9-663o 

1.2846 

9.4461 

786 

9.2160 

8.68i7 

o.  69-76 

+9.i8o3 

+8.9618 

—9.9466 

1.2844 

9.44-79 

787 

8.927o 

8.4049 

o.4372 

—8.7435 

—  9.7i3o 

+9-7977 

1.2835 

9.4691 

788 

8.8235 

8.3260 

0.6049 

+8.3oio 

—  9.6091 

—9.4666 

i.28i3 

9.4816 

789 

8.8369 

8.34o7 

o.5io3 

+8.4I75 

-9.  4558 

—9.6696 

1.2812 

9.4828 

79° 

8.9275 

8.43i5 

0.4334 

-8.7476 

-9.72l5 

+9-799° 

1.2812 

9.483o 

791 

8.9984 

8.5o93 

o.4ioi 

—8.8867 

—9,71x1 

+9.8666 

1.2806 

9.4891 

792 

8.8936 

8.4o66 

0.5277 

+8.6620 

—9.229-7 

-9.7465 

i.28o3 

9.4911 

793 

8.8o59 

8.35o8 

0.4986 

+8.0616 

—9.6660 

—9.2304 

1.2770 

9.6197 

794 

8.8693 

9.4o63 

0.9118 

+9.8676 

+9.4676 

—9-9729 

1.2768 

9.6216 

795 

9.o475 

8.6960 

0.3833 

—  8.9646 

—9.7169 

+9.8916 

1.2767 

9.6228 

796 

8.8072 

8.36i2 

o.5oo5 

+8.1224 

—  9.6601 

—  9.2890 

i  .2760 

9.6278 

797 

9.o473 

8.6o37 

o.38n 

—8.9647 

—  9.7i95 

+9.8909 

1.2768 

9.6299 

798 

8.9026 

8.4773 

o.5374 

+8.6979 

-9.0770 

—  9.7667 

1.2736 

9.6462 

799 

8.  -7918 

8.393o 

0.4870 

—  6.2719 

—9.  6386 

+7-4479 

1.2701 

9.6691 

800 

—8.9808 

—8.6890 

+o.3938 

-8.8637 

—9-7537 

+9.8498 

+  1.2691 

—9.6761 

432 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

ft.       TO.            S. 

s. 

o           /            7t 

» 

801 

4549 

Centauri,            e 

3 

i3  3o  a5.  18 

+3.735 

i4a  4s     3.5 

+  18.69 

802 

455a 

a5  Canum  Venat., 

5 

3o  46.67 

2.681 

62  56  26.8* 

18.60 

8o3 

4568 

83  Ursae  Majoris, 

5 

35     2.54* 

2.294 

34  33  28.4* 

18.41 

8o4 

4579 

i  Centauri,           i 

5 

87  10.64* 

3.389 

122     l6    69.6* 

18.47 

8o5 

458o 

Centauri, 

5 

37  11.59 

3.745 

i4o  4o  36.7 

18.12 

806 

4597 

4  Bootis,                r 

5 

4o    8.o4* 

2.856 

71  47  36.6* 

18.12 

807 

46oi 

Centauri,            v 

ft! 

4o  3i.88 

3.562 

i3o  56  i5.5 

18.26 

808 

4602 

Centauri,            p 

31 

4o  36.ii 

3.57o 

i3i  43  26.6 

18.26 

809 

46o3 

2  Centauri,            g 

5 

4o  46.47* 

3.452 

123    42       O.I* 

i8.3o 

810 

4607 

85  Ursse  Majoris,    77 

ii 

4i  37.27* 

2.352 

39  56  ii.  i* 

18.16 

811 

46i5 

5  Bootis,               v 

4 

42  14.59 

2.895 

73  27  20.4 

i8.o4 

812 

4623 

3  Centauri,           k 

41 

43  n.oo* 

3.443 

122     l4    54.6* 

18.17 

8i3 

4629 

4  Centauri,            h 

5 

44  35.42* 

3.43i 

121    ii     5.6* 

18.06 

814 

4638 

Centauri,           £ 

3 

46  12.67 

3.693 

i36  32  49.6 

18.06 

8x5 

4646 

10  Draconis,          .{ 

ft 

47    3.i4* 

i.755 

24  32     5.4* 

17.97 

816 

4648 

8  Bootis,                rj 

3 

47  32.54* 

2.862 

70  5o  64.2* 

18.24 

817 

4653 

Centauri,            <j> 

<* 

49  10.53 

3.6o8 

i3i   21   68.7 

18.01 

818 

4654 

Centauri,           v1 

5 

49  26.29 

3.654 

i  34     4     9.7 

18.06 

819 

4656 

9  Bootis, 

5 

49  43.  16 

2.746 

61  46   i3.2 

17.87 

820 

466o 

Apodis,              6 

5 

5o  54.43 

5.527 

166    4  i4'9 

18.06 

821 

4668 

Centauri,            v2 

5 

52  23.67 

3.695 

i  34  62  28.7 

17.81 

822 

4669 

Centauri,            (3 

i 

53  17.41 

4.i45 

i  49  38  44-6 

17.72 

828 

4672 

93  Virginis,             r 

41 

54     1.07 

3.o5o 

87  43  37.5 

17.69 

824 

468i 

Centauri,            % 

5 

56  54.  8z 

3.636 

i3o  27  3i  .9 

17.70 

825 

4685 

49  Hydrse,               ?r 

41 

57  So.Sg 

3.398 

116  67  23.9 

17.60 

826 

4686 

5  Centauri,            6 

2! 

57  5i.34* 

3.5o5 

126  37  49.4* 

18.09 

827 

4692 

Apodis,              77 

5 

59  45.17 

6.914 

170  17  43.o 

17.13 

828 

4696 

ii  Draconis,           a 

31 

i4    o  19.87* 

1.618 

24  54  21.4* 

I7.37 

829 

47o5 

Octantis,            d 

5 

3  29.22 

8.524 

172  58  23.8 

17.24 

83o 

4708 

5o  Hydrae, 

5 

4  11.19 

3.4i7 

116  33     7.3 

17.22 

83i 

4712 

Apodis,               e 

5 

4  39.94 

6.829 

169  24  20.9 

16.62 

882 

4716 

98  Virginis,            K 

4 

4  54.02* 

3.196 

99  34  23.5* 

i7-i4 

833 

4726 

17  Bootis,               K 

5 

8    6.4i* 

2.  169 

37  3o  24.0* 

17.0^ 

834 

4727 

99  Virginis,             t 

4 

8    9.33* 

3.142 

95   16  55.i* 

17.41 

835 

4729 

1  6  Bootis,                a 

i 

8  49.24* 

2.734 

70       2       3.9* 

18.93 

836 

4?32 

Ursae  Minoris,        4  5    N 

9  19.15 

-f-i.no 

19  5i  44-6* 

17.01 

837 

4733 

4  Ursae  Minoris, 

Var. 

9  30.89 

—  o.387 

ii  44  5i.6* 

16.96 

838 

4?34 

Lupi,                  L 

41 

9  49.72 

4-3.8o6 

i35  21  48.9 

17.12 

839 

474i 

19  Bootis,               a 

4 

10  40.70* 

2.288 

43  i3  i5.8* 

16.76 

84o 

4742 

21  Bootis,               « 

4 

10  5i  .01* 

2.i3o 

37  56  2i.3* 

16.80 

84  1 

4743 

100  Virginis,             A 

4 

II       O.OI* 

3.237 

102  4o  4o.8* 

i6.84 

842 

4745 

Centauri,            ip 

5 

ii  27.41 

3.619 

127  ii  33.9 

16.93 

843 

4759 

Centauri, 

5 

i3  48.86 

3.669 

128  49  2i.3 

16.73 

844 

4768 

Lupi,                  TI 

5 

i  6  32.09 

3.  812 

i34   32  24.0 

16.78 

845 

4770 

Lupi,                  r2 

5 

16  33.  81 

3.822 

i  34  4i  53.5 

16.71 

846 

4789 

23  Bootis,               6 

4 

20       5.32* 

2.o45 

37  27   i5.o* 

i6.85 

847 

4792 

io5  Virginis,             0 

5 

20    28.88 

3.090 

91    33     9.4 

16.42 

848 

48oi 

Lupi,                  o 

5 

22    32.78 

3.987 

i3g  47   17-2 

16.24 

849 

48o8 

25  Bootis,               p 

4 

25    21.85* 

2.690 

58  58     4.o* 

16.02 

85o 

48n 

Centauri,            77 

3 

i4  26    0.24 

+3.770 

i3i  29  44.6 

+  16.24 

TABLE    XXX. 


433 


CATALOGUE   OF    1500   STARS. 


No. 

Logarithms  of                                 j|                                  Logarithms  of 

a 

b 

c 

d 

of 

V 

c' 

d! 

801 

—  9.0067 

—8.6262 

+o.5735 

+8.9074 

+8.63i4 

—9.8669 

+    .2676 

—  9.5848 

802 

8.8869 

8.5o83 

0.4283 

—8.6670 

—  9.7625 

+9.7461 

.2672 

9.6864 

8o3 

9.0317 

8.6754 

0.3596 

—8.9474 

—  9.76i9 

+9.8772 

.2638 

9.6062 

8o4 

8.8566 

8.5ii2 

0.5338 

+8.5842 

—  9.  1676 

—9.6873 

.2610, 

9.6i43 

8o5 

8.9818 

8.6364 

o.573i 

+8.8703 

+8.6365 

—  9.8482 

.2619 

9.6x44 

806 

8.8o34 

8.4727 

o.46oi 

—8.2981 

-9.7359 

+  9.4619 

.2594 

9.6266 

807 

8.9026 

8.5738 

o.55i8 

+8.7189 

—8.6946 

-9.7732 

.259O 

9.6281 

808 

8.9077 

8.5794 

0.5536 

+8.7309 

-8.6128 

—9.7799 

.260.0 

9.6284 

809 

8.86o4 

8.5329 

o.5378 

+8.6o46 

—9^.0962 

—  9.7008 

.2688 

9.629.1 

810 

8.9723 

8.6489 

0.3776 

—8.8569 

-9.7836 

+9.84o5 

.2681 

9.  6325 

8n 

8.7975 

8.4772 

0.4623 

—  8.2520 

-9.73i5 

+9.4098 

.2676 

9.635o 

812 

8.85u 

8.5353 

0.5363 

+8.5783 

—  9.1297 

—9.6816 

.2666 

9.6387 

8x3 

8.8448 

8.5358 

0.5349 

+8.559o 

-9.i556 

—9.6673 

.2553 

9.644i 

8x4 

8.9381 

8.6368 

o.5684 

+8.7990 

+8.4i5o 

—9.8126 

.2538 

9.65o3 

8i5 

9.i564 

8.859i 

0.2435 

—9.1153 

-9-7707 

+9.9097 

.253o 

9.  6535 

8x6 

8.7990 

8.5o4o 

0.4565 

—  8.3i49 

—  9.7469 

+9.4663 

.2626 

9.6664 

817 

8.8973 

8.6100 

o.5576 

+8.7174 

—  8.356o 

—9.7689 

.2610 

9.6614 

8x8 

8.9160 

8.6298 

o.564i 

+8.7583 

+  7.8021 

—9.7908 

.2607 

9.6624 

819 

8.8271 

8.5423 

0.4377 

—  8.5o2o 

-9.7783 

+9.6231 

.2604 

9-6634 

820 

9.3894 

9.1101 

0.7465 

+9.3765 

+9.54o5 

—  9.934i 

.2493 

9.6677 

821 

8.9190 

8.6465 

0.5679 

+8.7676 

+8.3945 

—9-7941 

.2478 

9.673o 

822 

9.o65o 

8.7965 

o.6i84 

+9.0009 

+9-2790 

—  9.88o6 

.2460. 

9.6762 

828 

8.7681 

8.5o3o 

0.4836 

—7.3665 

—  9.6556 

+8.54a3 

.2461 

9.6787 

824 

8.8835 

8.63:3 

0.5596 

+8.6956 

—  8.i239 

—  9.753o 

.243  1 

9.688-7 

826 

8.  8100 

8.56i9 

o.53o3 

+8.45U 

—  9.2428 

—9.6810 

.2421 

9.69.18 

826 

8.8538 

8.6069 

0.5493 

+8.6191 

—8.8202 

—  9.7062 

.2421 

9.69i9 

827 

9.5349 

9.2953 

o.8434 

+9.5287 

+9  .6210, 

—9.9316 

,24OO 

9.6982 

828 

9.1367 

8.8995 

O.2Il5 

—  9.0943 

—9.8044 

+9.8948 

.2394 

9.7ooi 

829 

9.6700 

9.4466 

0.9363 

+9.6668 

+9.6692 

—9.9304 

.2359 

9.7IO2 

83o 

8.8062 

8.5847 

0.5334 

+8.4555 

—9.1978 

—9.  5832 

.235i 

9.7I24 

83i 

9.4918 

9.2734 

o.83o6 

+9-4843 

+9.6324 

—9.9248 

.2345 

9-7^9 

832 

8.7620 

8.5447 

o.5o35 

+  7.983o 

—9.5336 

—  9.i53o 

.2343 

9-7I47 

833 

8.9677 

8.7639 

o.33i8 

—8.8671 

—  9.8356 

+9.8277 

.23o5 

9.7245 

834 

8.7540 

8.55o4 

0.4964 

+7.7181 

—  9.5835 

—8.8923 

.23o5 

9.7247 

835 

8.7783 

8.5775 

0.4490 

—  8.3n6 

—9.7708 

+9.4608 

.220.7 

9..7267 

836 

9.2196 

9.0209 

+0.0370. 

—  9.1930 

—9.8132 

+9.9002 

.229I 

9.  -7282 

837 

9.4417 

9.2439 

—9.5707 

—9-4325 

-9.7873 

+9.9174 

.2288 

9.7288 

838 

8.9034 

8.7069 

4-o.5795 

+8.7557 

+8.8865 

-9.7786 

.2285 

9-7297 

839 

8.9i36 

8.7206 

0.3622 

-8.7761 

—9.8392 

+9.7878 

.2274 

9.7322 

84o 

8  .9602 

8.7679 

o.SSxa 

-8.8571 

—  9.8407 

+9.8219 

.2272 

9.7327 

84i 

8.7594 

8.5678 

0.6097 

+8.1008 

—  9.4839 

—  9.2662 

.2270 

9.7332 

842 

8.8469 

8.6572 

o.5589 

+8.6283 

—8.2480 

—9.7067 

.2265 

9.7345 

843 

8.8537 

8.6737 

o.5639 

+8.6509 

+  7.875i 

—9.7186 

.2236 

9-74i4 

844 

8.8889 

8.7200 

o..58o8 

+8.7-348 

+8.9268 

-9.7639 

.2201 

9-749I 

845 

8.8900 

8.7213 

o:.58i3 

+8.7.872 

+8.9360 

—  9-765o 

.2201 

9-749I 

846 

8.9532 

8.7987 

0:.3l58 

—8.  8529 

—9.8670 

+9.8i3o 

.2i55 

9.7588 

84? 

8.7368 

8.584o 

0.4902 

+7.1697 

—9.6214 

—8.3456 

.2l49 

9-7599 

848 

8.9239 

8.7794 

0.6012 

+8.8068 

+9.2167 

—9-7929 

.2122 

9.7654 

849 
85o 

8.7971 
—8.8546 

8.6638 
-8.  7238 

o.4i4o 
4-0.5768 

-8.5o93 
+8.6758 

—9.8309 
+8.8395 

+9.6i83        .2o83 
—9.7264'+    .2074 

9.7728 
—  9-7744 

EE 


434 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.  C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.        s. 

s. 

Q                1                  II 

85i 

4812 

27  Bootis,                y 

31 

l4    26      2.  2O* 

+  2.429 

5l       2       O.I* 

+  i5.98 

853 

4821 

Lupi,                  p 

5 

27   49.93 

+  3.998 

i38  46     7.9 

i6.25 

853 

4822 

5  Ursae  Minoris, 

4 

27    54.43* 

—  0.244 

i3  38  i4.3* 

i6.o5 

854 

4823 

28  Bootis,                a 

5 

28      9.02* 

+2.6i5 

59  36     3.7 

i5.89 

855 

483i 

Centauri,            a1 

4 

29    26.46 

4.018 

i5o  12  53.8 

i5.ii 

856 

4832 

Centauri,            a2 

i 

29    28.00 

4.018 

i5o  12  37.4 

i5.  ii 

857 

4833 

Apodis,               a 

41 

29    29.31 

7.009 

168  24     1.9 

i5.89 

858 

4835 

Circini,               a 

4 

3o  27.77 

4.739 

i54  19     0.9 

i6.i5 

859 

4839 

Lupi,                    a 

3 

3i   58.  96 

3.941 

i  36  44  26.2 

i5.88 

860 

4842 

Centauri, 

5 

32  39.59 

3.697 

127     8  45.7 

i5.83 

861 

4847 

29  Bootis,               TT 

a* 

33  4o.55 

2.816 

72  56     7.1 

i5.69 

862 

4849 

3o  Bootis,               £ 

31 

33  59.35* 

2.861 

75  37  3i.6* 

15.70 

863 

485o 

3  1  Bootis, 

5 

34  16.99 

2.947 

81   ii   36.9 

15.70 

864 

4852 

Centauri, 

5 

34  3o.o2 

3.637 

124  3i  26.3 

i5.93 

865 

4855 

107  Virginis,             ^ 

41 

35     9.62* 

3.i55 

95       O    12.2* 

i5.96 

866 

4864 

34  Bootis, 

4* 

36  49.91 

2.642 

62  49  53.6 

i5.53 

867 

4873 

35  Bootis,               o 

41 

38   14.67 

2.800 

72  23  49.6 

i5.48 

868 

4876 

36  Bootis,                e 

3 

38  26.14* 

2.622 

62  17  27.3* 

i5.45 

869 

4878 

109  Virginis, 

4 

38  4o.24 

3.029 

87  28  17.4 

i5.46 

870 

488o 

56  Hydrae, 

5 

39     o.oo 

3.478 

n5  27  2o.5* 

i-5.5i 

871 

4882 

57  Hydras, 

5 

39  11.59* 

3.483 

116     o  50.9* 

i5.48 

872 

4890 

7  Libras,                fj. 

5 

4l       6.22* 

3.279 

io3  3i    14.0* 

i5.3i 

873 

4891 

58  Hydras, 

5 

4i  29.44* 

3.5o3 

117   19  55.8* 

i5.32 

874 

4892 

Lupi,                  o 

5 

4i  52.47 

3.885 

i32   57     3.2 

i5.4i 

875 

4895 

9  Libras,                 a 

3 

42  35.3i* 

3.3o9 

io5  24  54.7* 

15.29 

876 

4go5 

37  Bootis,                | 

31 

44  28.28 

2.767 

70   16  28.4 

i5.25 

877 

4922 

1  5  Libras,                £2 

5 

48  38.i3* 

3.245 

100  48     4.4* 

14.90 

878 

4924 

Lupi,                  p 

3 

48  43.91 

3.891 

i32   3i   32.o 

i4.98 

879 

4928 

Centauri,            K 

3 

49  25.53 

+3.865 

i3i   29  52.6 

i4.8i 

880 

4g36 

7  Ursae  Minoris,  ft 

3 

5i    12.  o3* 

—  0.273 

i5   i3  53.8* 

14.78 

881 

4939 

19  Libras,                 6 

41 

52  57.84* 

+3.197 

97  55   i3.o* 

14.62 

882 

4948 

Lupi,                  TT 

5 

54  55.  81 

4.o39 

i36  27  36.4 

:4.63 

883 

4949 

Draconis, 

5 

55  12.83 

o.935 

23  28     9.4 

i4.42 

884 

495o 

20  Libras, 

31 

55  18.10* 

3.496 

n4  4i  19.7* 

i4.5o 

885 

495i 

no  Virginis, 

5 

55  19.57 

3.027 

87   18  67.7 

i4.46 

886 

4958 

42  Bootis,                (3 

3 

56  17.80* 

2.264 

49     o  55.2* 

i4.46 

887 

4969 

43  Bootis,               V 

5 

58     1.19 

2.572 

62  27   53.6* 

i4.3o 

888 

4970 

2  1   Libras,                 vl 

5 

58  16.19 

3.335 

io5  4o   16.8 

l4-32 

889 

49?3 

Lupi,                   /I 

5 

58  45.79 

3.996 

i34  4i   57.5 

i4.48 

890 

4974 

44  Bootis,                i2 

5 

58  5o.82* 

1.978 

4i  45  35.o* 

l4-22 

891 

4981 

45  Bootis,               c 

5 

i  5     o  42.75 

+  2.632 

64  32  36.o 

i4.3o 

892 

4982 

Ursae  Minoris, 

5 

o  44«o5 

—4.797 

6  52  24.6 

i4.i4 

893 

4986 

Lupi,                   K 

5 

i   31.97 

+4.n5 

i38     9  44.8 

14.19 

894 

4987 

Lupi,                  C 

4 

i  32.  3o 

4.25o 

i4i    3i   27.9 

i4.3i 

89'5 

5oo5 

Triang.  Aust.,    y 

3 

4  59.47 

5.456 

i58     7     9.5 

i3.92 

896 

Son 

Circini,               ft 

5 

5  48.85 

4.6i3 

i48  i4    4.8 

i3.85 

897 

5o28 

Lupi,                  fj. 

5 

8     7.65 

4.127 

137  19     6.9 

i3.85 

898 

5o3i 

48  Bootis,               x 

5 

8  12.96 

2.5o8 

60   16  35.8 

i3.65 

899 

5o32 

2  Lupi, 

41 

8  42.96* 

3.63i 

119  35  33.9* 

13.69 

900 

5o34 

27  Libras,                 ft 

2! 

i5     8  56.44* 

+  3.220 

98  49  33.!* 

+  I3.62 

TABLE   XXX, 


435 


CATALOGUE   OF   1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

b'        \        c' 

d' 

85i 

—8.8383 

-8,7077 

+o.385i 

—  8,6369 

—  9.8522 

+9.703-7 

+  i  .2073 

-9.7746 

862 

8.9076 

8.784o 

+0.6007 

+8.7838 

+  9.2206 

-9.7788 

1.2048 

9.7790 

853 

9.3539 

9.23o6 

-9.3876 

—  9.34i5 

—  9.8368 

+9.8901 

I  .2047 

9.7792 

854 

8.79o3 

8.6680 

+0.4147 

—  8.4945 

—  9.83i9 

+9.6063 

1.2044 

9.7798 

855 

9.0281 

8,9108 

0.6621 

+8.9665 

+9.4929 

—9.8387 

1.2026 

9.783i 

856 

9.0280 

8  .9io9 

0.6621 

+8.9664 

+9.4929 

—9.3387 

1,2026 

9.783i 

857 

9.4208 

9.3o38 

0.8466 

+9.4118 

+9.7o53 

—9.8913 

1.2024 

9.7832 

858 

9.0868 

8,9726 

o.6783 

+9.o4o6 

+9.6687 

—  9.8536 

I  .2010 

9.7856 

859 

8.8846 

8  .7773 

0,6961 

+8.7469 

+9.1787 

—9.7689 

1.1988 

9.7893 

860 

8.  8180 

8,7133 

0.6678 

+8.5989 

+8.4346 

-9.6766 

1.1978 

9-79°9 

861 

8,7375 

8.6368 

0.4496 

—  8.2060 

—9.7762 

+9.36i6 

1.1963 

9.7934 

862 

8.73i3 

8.63i8 

o,456o 

—8.1262 

—9.7696 

+9.2886 

1.1968 

863 

8.7222 

8,6239 

0.4686 

—  7.9o7i 

—9.7196 

+9.0781 

1.1963 

9.7948 

864 

8.8008 

8.7034 

0.6617 

+8.5542 

—7.6682 

—  9.6462 

I  .  1960 

9.7963 

865 

8.7i74 

8.6226 

0.4976 

+7.6579 

-9,6772 

—9.  8323 

i.i94o 

9.7969 

866 

8.7639 

8.  6766 

0.4211 

—8.4234 

—  9.8291 

+9.6488 

1.1914 

9.8008 

867 

8.73i8 

8,6488 

0.4472 

—8.2124 

—9.7826 

+9.3676 

1.1892 

9.8041 

868 

8.7635 

8.68x3 

o.4i88 

—  8.43io 

—9.833o 

+9.5542 

1.1889 

9.8046 

869 

8.7io7 

8.6294 

0.4819 

—7.3553 

—9.6644 

+  8.53io 

1.1886 

9.8061 

87o 

8.754i 

8.,674i 

o.54i3 

+8.3874 

—  9  .0682 

—9.6191 

1.1881 

9.8068 

871 

8.  7668 

8.6766 

0.5426 

+8.3979 

—  9.o382 

—  9.6276 

1.1878 

9.8062 

872 

8.7186 

8.6467 

0.6167 

+8.o874 

-9.4327 

—9/2613 

i.i847 

9  .  8  1  06 

873 

8.7672 

8.6867 

0.5463 

+8".4i9i 

—  8.945o 

—  9.5438 

i.i84i 

9.8ii4 

874 

8.8407 

8.7717 

o.5887 

+8.674i 

+9.o955 

—  9,7i46 

i.i835 

9.8123 

875 

8-7199 

8.6537 

0.6199 

+8.i445 

—  9.3897 

—  9,3o47 

1.1823 

9.8139 

876 

8.7272 

8.  6681 

0,4402 

—8.2555 

—9-7997 

+9.4o53 

1.1792 

9.8180 

877 

8.7017 

8.6585 

0.6108 

+  7-9744 

—9.4778 

—  9.  1428 

i  .  1722 

9.8269 

878 

8.8263 

8.7835 

0.6904 

+8.6662 

+9.1274 

—  9.6997 

i  .  1720 

9.8271 

879 

8.8181 

8-7779 

+o.5874 

+8,6393 

+9.0842 

—9.6899 

1.1709 

9.8286 

880 

9.2699 

9.2366 

—9.4242 

—  9.2544 

-9.  8863 

+9.8600 

1.1678 

9.8322 

881 

8.69o5 

8.6638 

+o.5o49 

+7.8297 

—  9,5256 

—  9.0016 

i.i646 

9.8357 

882 

8.8447 

8.8264 

0.6067 

+8.7o49 

+  9.3o58 

—9.7192 

i  .  1611 

9.8397 

883 

9.o82i 

9.o64o 

9-9727 

—  9,o446 

—  9,9o73 

+9.8209 

i.  1606 

9  .  8402 

884 

8.7237 

8.7o59 

0.5436 

+8.3446 

—  9,02l6 

—9.4791 

i  .  1  60^ 

9  .  84o4 

885 

8.6826 

8.6648 

0.4810 

—  7,353o 

—  9,6683 

+8.6286 

1.1602 

9.8406 

886 

8.8024 

8.7884 

0.3546 

—  8.6i92 

—  9.8942 

+9,6732 

1.1686 

9.8424 

887 

8.7293 

8.7218 

0.4120 

—  8.3942 

—  9.85o6 

+9.6181 

i.i554 

9.8457 

888 

8.693i 

8.6865 

0.5229 

+8.1247 

-9.3583 

—9.2843 

1.1660 

9.8462 

889 

8.8240 

8.8i93 

O.6O22 

+8.6712 

+  9.27OO 

—9,6990 

i.i54o 

9.8471^ 

89o 

8.8621 

8.8477 

o,3o47 

—8.7248 

—  9.9106 

+9.7244 

i.i539 

9-8473 

891 

8.7164 

8.7191 

+0.4182 

-8.3497 

—9.8431 

+9.48i4 

i.i5o4 

9.8608 

892 

9.594o 

9.6968 

—  0.6810 

—  9.59o9 

—  9.8786 

+9.8460 

i.i5o3 

9.8609 

893 

8.8463 

8.8621 

+o.6i63 

+8.7186 

+9.38i7 

—9.7187 

i.  i486 

9.8624 

894 

8.8765 

8.8824 

o.63o2 

+8.7702 

+9.4583 

—9.7402 

i.i488 

9.8624 

895 

9.0924 

9.1113 

o.7384 

+9.o599 

+9-7°64 

—9.8074 

I.I420 

9.8687 

896 

8.94o7 

8.9628 

0.6661 

+8.8708 

+9.5886 

—9.7677 

i.i4o4 

9.8602 

897 

8.8263 

8.857i 

0.6161 

+8.6926 

+9.3869 

—9.6999 

i.i357 

9.  8644 

898 

8.7186 

8.7497 

o.4ooo 

—  8.4i38 

—  9.869i 

+9.6287 

i.i356 

9.8645 

899 

8.7i69 

8.75oo 

o.5596 

+8.4io5 

—  8.1818 

—  9.6269 

i.i345 

9.8654 

900 

—  8.66o9 

—  8.6949 

+o.5o83 

+  7.8468 

—  9.5ooo 

—9.0178 

+  i.i34i 

—  9.8658 

436 


TABLE   XXX, 


CATALOGUE    OF    1500    STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.. 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.         s. 

t. 

o                    // 

// 

901 

5o36 

49  Bootis,               6 

31 

i5     9  27.44* 

4-2.421 

56     7  21.8* 

+  i3.67 

902 

5o46 

Lupi,                  d 

4 

ii   32.  5o 

3.898 

i3o     6     4.1 

13.65 

9o3 

5o49 

Lupi,                  v1 

5 

ii  42.44 

4.i38 

137  22  36.i 

13.67 

904 

5o54 

Lupi,                  01 

5 

12  18.66 

3.791 

126  42  44-  9 

13.46 

906 

5o56 

Lupi,                 e 

41 

12  3o.65 

4.028 

i  34     8  44.4 

1  3.  5  7 

906 

5o6o 

Lupi,                 03 

5 

i3  35.42 

+3.802 

126  18  58.8 

18.39 

907 

6079 

1  1   Ursae  Minoris, 

5 

17  16.32* 

—0.086 

17  37  53.5 

1  3.  06 

908 

5o84 

5  1   Bootis,               fj. 

4 

18  49.42* 

+2.267 

62.     5  39.0 

12.89 

909 

5o89 

32  Librae,                £: 

4 

19  48.27* 

+3.374 

i  06  ii   22.9* 

12.9^ 

910 

5o94 

1  3  Ursae  Minoris,   y 

3| 

21     o.38* 

—  0.146 

17  37  56.o* 

12.77 

911 

5o97 

12  Draconis,           t 

3- 

21   36.oo* 

+I.332 

3o  3o  24.8* 

12.76 

912 

5o98 

3  Coronae  Bor.,    J3 

4 

21  38.99 

2.48o 

60    22    27.3* 

12.71 

9i3 

5io3 

Triang.  Aust.,   e 

5 

23       3.22 

5.356 

166  48  22.5 

12.98 

9i4 

5n8 

Lupi,                  7 

3 

25     9.82 

3  ..963 

i3o  39  29.7 

12.71 

916 

5i25 

37  Librae, 

4 

25  59.19* 

3,268 

99  32  46.7* 

12.72 

916 

5i3i 

4  Coronae  Bor.,    0 

41 

26  62.95 

2.4i5 

58     7  62.6 

12.4^ 

917 

5i34 

38  Librae,                y 

41 

27     8.48* 

3.  .344 

104  17     6.7* 

12.39 

918 

5i35 

1  3  Serpentis,           6 

3 

27  38.64 

2.866 

78    67    20.3 

12.32 

919 

5i38 

3  9  Librae, 

4 

27  55.82* 

3.624 

117  38     1.5* 

12.32 

920 

5i39 

Lupi,                  i 

5 

27  5&.i8 

4.018 

182    4  16.9 

12.  49 

921 

5i43 

5  Coronae  Bor.,     a 

2l 

28    20.25* 

2.539 

62  46  39.0* 

12.  39 

922 

5i5i 

4o  Librae, 

41 

29    27.28* 

3.662 

119  16  47.7* 

12.25 

923 

5i55 

6  Coronae  Bor.,    p 

5 

29    44.71* 

2.  .200 

5o  29  20.8* 

12  ,2: 

924 

5i65 

Lupi, 

5 

3o  53.87 

4-090 

i34     9  32.  9 

12.49 

925 

6176 

43  Librae,                K 

5 

33  i8.75* 

3.445 

109  ti    18.6* 

12.  06 

926 

6178 

7  Coronae  Bor.,     f 

5 

33  44.09 

2.271 

62  62  28.2 

I2.o5 

927 

6187 

2  1   Serpentis,          i 

5 

34  51.89 

2.670 

69  5o  34.8 

11.89 

928 

5i9o 

44  Librae,                 77 

41 

35  38.48* 

+3.37i 

106   ii  27.1* 

11.86 

929 

5i9i 

1  5  Ursae  Minoris,   0 

5 

35  58.i5* 

-1.976 

12       9     II.9* 

11.79 

93o 

5l92 

8  Coronae  Bor.,     y 

5 

36  26.63 

+2.619 

63  i3  3i.5 

11.68 

93i 

6196 

24  Serpentis,..          a 

2l 

36  52.  92* 

2.963 

83     5  56.o* 

11.67 

932 

6214 

27  Serpentis,          /I 

41 

39  10.  i5* 

2-912 

82     10    23.  9* 

1  1.  60 

933 

6216 

28  Serpentis,          /3 

31 

39  16.99 

2.767 

74     616.2 

H.56 

934 

6224 

Triang.  Aust.,.  K 

5 

4o  46.23 

5.836 

168     8  62.2 

ii.46 

935 

6227 

5  Lupi,                  x 

4 

4i  26.61* 

3.789 

123     9  56.5* 

n.46 

936 

523o 

32  Serpentis,          ft 

3-1 

4i  47.94 

3.128 

92  58     0.4 

H.38 

937 

5232 

i  Scorpii,              d 

5 

4i  58.  o5* 

3.592 

n5   17  26.2* 

ii.  4o 

938 

5233 

Triang.  Aust.,   (3 

3 

4  1  59.io 

5.2o5 

162  67  36.8 

11.73 

o39 

5234 

35  Serpentis,          AC 

4 

4i  59.54 

2.700 

71    23  28.4 

ii  .41 

|4o 

5244 

10  Coronae  Bor.,     6 

41 

43  i8.29 

2.5l2 

63  28     6.7 

ii.  3o 

94i 

5245 

37  Serpentis,          e 

3 

43  20.67 

2.989 

85     4     3.3 

II.  21 

94s 

525o 

2  Scorpii,              A 

5 

44  37.01* 

3'.  587 

n4  62  29.1* 

11.19 

943 

525i 

45  Librae,                /I 

4 

44  38.i5* 

3.471 

109  42  60.7* 

II  .  I9 

944 

5252 

38   Serpentis,.          p 

41 

44'  4'o.8o 

2.635 

68  34     1.4 

II.  II 

945 

5257 

46  Librae,                6. 

41 

45  17.47* 

3.4io 

06  17     5.3* 

10.  99 

946 

5259 

n   Coronae  Bor.,     K 

5 

45  34.78 

2.266 

53  62  25.6 

11.43 

947 

5268 

Lupi,                  <f 

41 

47  18.96 

3.817 

23  3i    16.4 

10.  98 

948 

5272 

5  Scorpii,              p 

4 

47  37.93* 

3.689 

18  46  17.6* 

10.  97 

949 

5279 

Draconis, 

5 

48  47-63* 

i.4oi 

33  43  42.6* 

10.86 

95o 

5284 

4  1  Serpentis,           y 

3 

5  49  3i  .72* 

+2.769 

73  5o  4o.4 

-12.  o5 

TABLE   XXX. 


437 


CATALOGUE   OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

b' 

c'                d' 

9OI 

—  8.7355 

-8.7714 

+0.3821 

—8.4817 

—9.8867 

+9.577o 

+  i.i33o 

—9.8667 

902 

8.7668 

8.8106 

o.59i7 

+8.6768 

+  9.l64l 

—  9.6355 

1.1287 

9.8703 

908 

8.8193 

8/8638 

0.6179 

+8.6861 

+  9-4019 

—  9.6929 

1.1284 

9.87o6 

904 

8.7392 

8,7860 

0.6782 

+8.5o54 

+8.9i33 

—  9.6911 

i  .  1271 

9.8-716 

9o5 

8.7926 

8.84oo 

0.606  i 

+8.6354 

+9.3i64 

—  9.6673 

1.1267 

9-87i9 

906 

8.739-8 

8.79i5 

+o.58o4 

+8.5i23 

+8.9694 

—9.6946 

i.  1244 

9.8738 

907 

9.i563 

9.2226 

—9.0770 

—9.1369 

-9.93i5 

+9-7933 

i.n64 

9-8799 

908 

8.7375 

8.8092 

+o.3573 

—8.6260 

—  9.9090 

+9.6991 

i  .1129 

9.8824 

909 

8.65oo 

8.7254 

+0.6273 

+8.0963 

—9.3o84 

—9.  2538 

I.no7 

9.8839 

910 

9.1484 

9.2286 

—9.2138 

—  9.  1276 

—9.9366 

+9.7849 

I  .  1080 

9.8869 

911 

8.9228 

9.0061 

+  O.I2I3 

—8.858i 

—  9.9466 

+9-7397 

1.1066 

9.8868 

912 

8.6891 

8.7716 

o.3953 

—8.3831 

—9.8798 

+9.4983 

i.io65 

9.8869 

9i3 

9.0124 

9.  ioo4 

0.7304 

+8.9726 

+9.7232 

—  9.7612 

i.io33 

9.8890 

914 

8.7400 

8.8362 

0.6984 

+8.6640 

+9.2533 

—  9.6101 

1.0984 

9.8923 

9i5 

8.624^ 

8.  7235 

o.5n4 

+  7.8439 

—  9-4745 

—  9.0139 

i  .0964 

9.8935 

916 

8.68a9 

8-7897 

0.3835 

—8.4096 

—9.8938 

+9-5i47 

1.0943 

9.8949 

917 

8.6290 

8.7328 

0.5235 

+  8.0212 

—9.3553 

—9.1837 

i  .0936 

9.8963 

918 

8.6222 

•8.7280 

0.4672 

—  7.9046 

—  9.76io 

+9.0726 

i  .0924 

9.8960 

919 

8.6660 

8.7729 

0.6690 

+8.3324 

—8.2810 

-9.4559 

1.0917 

9.8964 

920 

8.7427 

•8.8498 

o.6o44 

+8.5688 

+9.3i37 

—9.6166 

i  .0916 

9.8966 

921 

8.6634 

•8.7719 

0.4028 

—8.3238 

-9.8732 

+9.4489 

i  .O9O7 

9.897o 

922 

8.6691 

8.7820 

o.564o 

+8.3584 

+  7.9445 

—9.4752 

i.  0880 

9.8987 

923 

8.7217 

8.8357 

o.34i8 

—8.5253 

—9.9238 

+9.6887 

1.0873 

9.899i 

924 

8.75o5 

•8.8688 

o.6i32 

+8.5935 

+9.3856 

—  9.6264 

i.o846 

9.9oo7 

925 

8.6249 

8.  -7629 

0.5370 

+8.i4i6 

—  9.  1662 

—  9.2929 

1.0784 

9.9o42 

926 

8.6974 

8.8271 

o.3537 

—8.4781 

—  9.9196 

+9.5558 

1.6778 

9.9o48 

927 

8.6235 

•8.7578 

0.4273 

—8.1608 

—9.  8358 

+9.3096 

i  .  0744 

9.9064 

928 

8.6095 

•8.7468 

+0.6269 

+8.0278 

—  9.3162 

—9.1886 

i  .0724 

9.9075 

929 

9.2699 

9.4o85 

—  0.2904 

—  9.2600 

—  9.9456 

+9.7694 

i  .0716 

-9.9079 

g3o 

8.64i2 

8.7817 

+  O.4O2I 

—8.2949 

—  9.8765 

+9.4217 

i  .0703 

9.9086 

93  1 

8.5940 

8.7362 

0.4683 

-7.6737 

—9.7229 

+8.8467 

i  .0691 

•9.9092 

932 

8.5888 

8  .  74o2 

0.4654 

—7.7.229 

-9.7334 

+8.8960 

i.o63i 

9.9123 

933 

8.6oi4 

8.7532 

o.44o8 

—8.0390 

—  9.8o7i 

+9:1981 

1.0628 

9.9124 

934 

9  .  0096 

9.  i675 

0.7640 

+8.9772 

+9.7786 

—9.7241 

1.0687 

9.9144 

935 

8.6558 

8.8x64 

0.6786 

+8.3939 

+8.9385 

—9.4927 

1.0669 

9.9i53 

936 

8.5782 

8  .  7402 

0.4962 

+7'.  292  1 

—  9.592O 

—8.4676 

1.0669 

9«9l57 

937 

8.6209 

8.7836 

0.6662 

+8.2616 

—  8.6o64 

—9.3839 

i.o554 

9.9i6o 

938 

8.9194 

9.0823 

o..7i85 

+8.8692 

+9.7297 

—9.7029 

i.o554 

9.9i6o 

939 

8.6oo4 

8.7632 

o.43i3 

—8.1043 

—9.8289 

+9.2671 

i.o554 

9.9i6o 

94o 

8.6218 

8.7899 

o.4oi  i 

—8.2718 

—9.8-796 

+9.3995 

i  .0617 

9.9i77 

941 

8.5749 

6.7433 

o..4735 

—7.6093 

—  9.  -7020 

+8.6838; 

i  .0616 

9.9i77 

942 

8.6120 

8.7866 

0.5546 

+8.2359 

—8.6464 

—9.3697 

i  .o48o 

9.9194 

943 

8.5969 

8.7696 

o.54oi 

+8.1240 

—  9.  1082 

—9.2738 

i.o48o 

9.9194 

944 

8.6oo7 

8.7745 

O.42O7 

—8.1635 

—9.8498 

+9.3o84 

1.0479 

9.9194 

945 

8.5856 

8.7619 

0.5309 

+8.o334 

—  9.2629 

—9.1917 

i  .  o46i 

9.9202* 

946 

8.6597 

8.8372 

0.3537 

-8.43o3 

—  9.9261 

+9.5i36 

i.o453 

9.9206 

94? 

8.64io 

8.826-7 

o.58n 

+8.383i 

+9.0022 

—9.4802 

I.o4o3 

9.9228 

948 

8.6i83 

8.  8o43 

0.5665 

+8.3007 

+8.3579 

—9.4196 

1.0394 

9.9231 

949 

8.8:32 

9.0040 

O.  l422 

—  8.733i 

—9.9-732 

+9.6537 

i.o36o 

9.9246 

960 

—  8.573o 

—8.  -7669 

+0.4384 

—8.0176 

-9.8i37 

+9.1761 

+  i.o338 

—  9.9266 

438 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.,     Annual 
Jan.  1,  1850.        Variation. 

h.     m.        s. 

s. 

0               /                  M 

•' 

95i 

5285 

1  6  Ursae  Minoris,   f 

4 

i5  49  31.76* 

—  2.3i6 

ii  44  48.o* 

+  10.81 

962 

5289 

6  Scorpii,              TT 

3i 

49  47-12* 

+3.6i5 

ii5  4o  39.2* 

10.85 

953 

5290 

48  Librae, 

41 

49   47.74* 

3.349 

io3  5o  32.5* 

10.78 

954 

5292 

Lupi,                  7? 

41 

5o  11.69 

3.946 

127  57  44.5 

10.87 

955 

53o2 

1  3  Coronae  Bor.,     e 

4* 

5i  22.73* 

2.487 

62  4i     4.7* 

10.70 

956 

53o3 

7  Scorpii,              6 

3 

5i  28.  3i* 

3.535 

112     II    24.8* 

10.68 

957 

5322 

44  Serpentis,           TT 

4| 

55  5o.i4 

2.58i 

66  46  31.7 

10.26 

958 

5323 

Normae,              <5 

5 

55  54.22 

4.198 

i  34  45  4i.i 

10.49 

959 

5324 

5  1   Librae, 

4f 

56     7.67* 

3.291 

100  57  18.0 

10.  3/ 

960 

5329 

8  Scorpii,              /31 

2 

56  43.  3i* 

3.478 

io9  s3  25.3* 

10.  3o 

961 

533i 

Lupi,                  6 

ii 

56  45.38* 

3.921 

126    23    21.3* 

10.35 

962 

5337 

9   Scorpii,              w1 

4i 

58     2.5i 

3.5o2 

no   i5  27.2 

10.  17 

963 

5338 

6  Herculis,            v 

5 

58     7.43* 

1.861 

43  32  4o.5* 

10.  2^! 

964 

5342 

10  Scorpii,              uz 

41 

58  36.94* 

3.5o8 

no  27  3i  .0* 

10.  i5 

965 

534? 

Scorpii, 

5 

58  59.55 

3.642 

ii5  55  i3.3 

10.17 

966 

5348 

1  3  Draconis,           6 

3 

59       5.22* 

i  .  124 

3i      i  58.6* 

9.78 

967 

5375 

Triang.  Aust.,   6 

41 

16     i  49.66 

5.364 

i53   17  41-7 

9.88 

968 

538i 

1  3   Scorpii,              c2 

5 

3     4.5o* 

3.689 

117  3!   55.i* 

9.80 

969 

5382 

1  4  Scorpii,              v 

4 

3  17.04* 

3.478 

109     3  58.  9* 

9.75 

97° 

5386 

1  5  Scorpii,             ^ 

5 

3  48.53* 

3.271 

99  4o  16.8 

9.78 

971 

5388 

ii   Herculis,           ^ 

5 

4     2.i5 

1.869 

44  4o  10.8 

9-7° 

972 

54o6 

Draconis, 

5 

5  55.91* 

o.  127 

21  47  39.6* 

9.5o 

973 

54i4 

i   Ophiuchi,           d 

3 

6  29.31* 

3.i38 

93  18  i4.5* 

9.6^ 

974 

5420 

1  8  Scorpii, 

5 

7  28.  3o* 

3.25o 

97  58     3.9 

9.96 

976 

5425 

Normae,              7* 

5 

8  38.46 

4.455 

i  §9  46  55.2 

9.  52 

976 

5437 

2  Ophiuchi,           t 

3 

10    23.  4l* 

3.i68 

94    19    22.2* 

9.19 

977 

5439 

Apodis,              7 

5 

10  36.26 

8.782 

168   33     7.8 

10.34 

978 

5447 

20  Scorpii,              a 

4 

12     4.73* 

3.634 

ii5  i3  39.8* 

9.10 

979 

5456 

5o  Serpentis,          a 

5 

i4  28.83 

3.o35 

88  36  49.4 

8.84 

980 

5459 

Draconis, 

5 

i4  45.56* 

+o.983 

29  5i  47.5* 

8.89 

981 

5462 

19  Ursae  Minoris, 

5 

i5     9.79* 

—  1.863 

i3  44  49.o* 

8.86 

982 

5463 

22   Herculis,            r 

4 

:5  i3.93* 

-M-799 

43   19  37.6* 

8.83 

983 

5466 

20  Herculis,            7 

31 

i5  18.29 

2.645 

70  29  28.4 

8.80 

984 

5467 

4  Ophiuchi,           i// 

5 

i5  19.86* 

3.5oi 

109  4o  53.4* 

8.90 

985 

5473 

19  Coronae  Bor.,     £ 

5 

16  15.27 

2.34o 

58  45  22.3 

8.63 

986 

5477 

5   Ophiuchi,           p 

5 

16  SS.go 

3.586 

ii3     5  44.8 

8.72 

987 

5479 

20   Coronae  Bor.,     vl 

5 

16  42.67* 

2.261 

55  5o  41.2* 

8.82 

988 

548o 

21    Coronae  Bor.r    r2 

5 

16  5o.23* 

2.269 

55  56  4o.9* 

8.75 

989 

5489 

7  Ophiuchi,           % 

5 

18  20.16* 

3.468 

108     6  4o.i* 

8.60 

99° 

5490 

24  Herculis,           w 

5 

18  29.44 

2.752 

75  37     2.5 

8.62 

991 

5494 

Ophiuchi, 

5 

19  37.94 

3.225 

97  i4  5i  .0 

8.5i 

992 

5495 

3  Ophiuchi,           v 

5 

19  41.78* 

3.254 

98     i-  55.3* 

8.5o 

998 

5496 

25  Herculis, 

5 

20       3.57 

2.i34 

52  i5  4o.8 

8.4y 

994 

5498 

21   Scorpii,              a 

i 

20  13.07* 

3.668 

116     5  38.  9* 

8.49 

995 

55o2 

Draconis, 

5 

21        8.75* 

i.3oo 

34  27     6.9* 

8.39 

996 

55o8 

Scorpii, 

| 

21   35.  4o* 

3.899 

124    22    24.8* 

8.43 

997 

55io 

Apodis,              p 

5 

21    49.4l 

+8.3i3 

167    II    26.5 

8.63 

998 

55n 

21   Ursae  Minoris,    77 

5 

21     57.21* 

—  i.839 

i3  54     4-8* 

8.12 

999 

55i2 

1  4  Draconis,           ij 

3 

21   58.36* 

+0.820 

28     8  42.4* 

8.24 

1000 

55i6 

8  Ophiuchi,           <p 

41 

16  22  33.  61* 

+3.43o 

106  16  5o.9* 

+  8.3o 

TABLE    XXX. 


439 


CATALOGUE    OF    1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b        |         c                  d 

a' 

b' 

c' 

df 

95i 

—  9.2468 

—  9.44o6 

—  0.3701 

—9.2376 

—  9.96o5 

+9.7224 

+    .o338 

—  9.9255 

952 

8.5999 

8.7948 

+o.5578 

+8.2367 

—  8.4i33 

—9.3676 

.o33i 

9.9258 

953 

8.5675 

8.7625 

0.5248  +7<9464 

—  9.34i8 

—9.1097 

.o33o 

9-9258 

954 

8.6568 

8.8534 

o.5967+8.4458 

+9.248o 

—  9.5i86 

.0319 

9.9263 

955 

8.6oi3 

8.8o29 

o.3955 

—8.2631 

—9.8888 

+9.3878 

.0283 

9.9277 

966 

8.583i 

8.785i 

o.548i 

+8.i6o3 

-8.9i38 

—  9.3029 

.0280 

9.9278 

957 

8.573o 

8.7935 

o.4n5 

-8.1688 

-9.8673 

+9.3082 

.oi46 

9.9329 

968 

8.6848 

8.9o56 

0.6240 

+8.5324 

+9.4703 

—  9.5598 

.0144 

9.933o 

959 

8.5433 

8.765i 

o.5i75 

+7.8222 

—9.4203 

—  8.99o3 

.oi37 

9.9332 

960 

8.5588 

8.7832 

o.54o9 

+8.0800 

—9.0941 

—  9.23o7 

.0118 

9.9339 

961 

8.6276 

8.8520 

o.5929 

+8.4008 

+9.2047 

—  9.4827 

.OII7 

9.9339 

962 

8.557o 

8.787o 

0.5436 

+8.0964 

—  9.0362 

—  9.2447 

.0076 

9.9354 

9G3 

8.69o8 

8.92I2 

O.269I 

—  8.55n 

—9.9655 

+9.5653 

.oo73 

9.9355 

964 

8.5557 

8.7882 

0.5442 

+8.0992 

—  9.0212 

—  9.247o 

.0067 

9.936o 

965 

.  8.5722 

8.8o64 

o.56oi 

+8.2128 

—8.1038 

—  9.3429 

.oo45 

9.9364 

966 

8.8i36 

9.o482 

o.o6o9 

—  8.7465 

—9.9841 

+9.6349 

I  .  0042 

9.9365 

967 

8.8642 

9.  i  i  08 

o.73io 

+  8.8i52 

+9.765i 

—9.6439 

0.9951 

9.9395 

968 

8.5648 

8.8170 

o.5657 

+  8.2297 

+8.2695 

—9-3536 

0.9909 

9.94o8 

969 

8.5364 

8.  7895 

o.54o8 

+8.o5o5 

—  9.0969 

—  9.2021 

0.9902 

9.94n 

970 

8.5i64 

8.7718 

o.5i45 

+7-?4i7 

-9.4486 

—  8.9n5 

o.9885 

9.94i6 

971 

8.6624 

8.9188 

o.2759 

—  8.5i44 

—  9.9672 

+9-5374 

0.98-7-7 

9.94i9 

972 

8.933i 

9.  1981 

9.  1222 

—  8.9009 

—9.9899 

+9.6467 

0.9811 

9.9438 

973 

8.5oi6 

8.7690 

o.4967 

+7.2622 

—9.5826 

—8.4376 

o.9792 

9  .  9444 

974 

8.5oi6 

8.7735 

o.5ioo 

+  7.6434 

—9.4880 

—  8.8i53 

0.9757 

9.9454 

975 

8.6832 

8.9605 

o.65o2 

+8.566i 

+9.5966 

—  9.5522 

0.9715 

9.9466 

976 

8.488i 

8.7735 

o.4997 

+  7.3653 

-9.  5632 

—8.5402 

0.9652 

9.9483 

977 

9.i884 

9.4747 

o.95n 

+9.i797 

+9-8945 

—  9.6535 

0.9644 

9.9485 

978 

8.5242 

8.8i74 

o.56o2 

+8.i538 

—  8.  10-72 

—9.2864 

0.9590 

9.95oo 

979 

8.4717 

8.7763 

o.483i 

—6.8555 

—9.6585 

+8.o3i4 

0.9499 

9.9522 

980 

8.7734 

9.o792 

4-9.9927 

—  8.7n5 

—9-9972 

+9-5848 

0.9489 

9.9525 

981 

9.o932 

9  .4009 

—  o.263o 

—  9.o8o6 

—  9.9888 

+9.6326 

0.9474 

9.9529 

982 

8.6323 

8.9404 

+o.255o 

—  8.494i 

—  9.978o 

+9.5o66 

0.9470 

9.9530 

988 

8.494: 

8.8026 

0.4225 

—  8.oi78 

—  9.85o5 

+9.1682 

0.9468 

9.9530 

984 

8.4945 

8.8o3i 

o.544o 

+8.0218 

—9.0265 

-9.1718 

0.9467 

9.9530 

985 

8.5328 

8.8459 

o.3694 

—  8.2477 

—9.9229 

+9.3558 

0.9431 

9.9539 

986 

8.4997 

8.8i44 

0.5543 

+8.o933 

—8.6646 

—  9.233i 

0.9417 

9.9542 

987 

8.5452 

8.86o5 

o.353o 

—  8.2945 

—9.9368 

+9.3884 

o.94i3 

9.9543 

988 

8.5442 

8.  8601 

0.3535 

—  8.2924 

—9.9364 

+9.3867 

o.94o8 

9.9544 

989 

8.4786 

8.  8018 

o.5398 

+7.9712 

—  9.  1196 

—  9.  1252 

o.9349 

9.9558 

990 

8.4698 

8.7937 

o.44io 

—  7.8649 

—  9.8094 

+  9.0272 

o.9343 

9.9559 

991 

8.4548 

8.7843 

o.5o85 

+7.5557 

—  9.5ooo 

—  8.7284 

o.9297 

9.957o 

992 

8.4554 

8.7852 

o.5io8 

+7.6oo6 

—9.4817 

—  8.7724 

0.9294 

9.957o 

993 

8.55i5 

8.8832 

0.3289 

—8.3383 

—9.9533 

+9.4i25 

0.9279 

9.9573 

994 

8.4957 

8.8281 

o.564o 

+8.i39o 

+7.9590 

—  9.2684 

O.9273 

9.9575 

995 

8.6926 

9.O296 

o.n4o 

—8.6088 

—9.9976 

+9.5375 

0.9235 

9.9583 

996 

8.5267 

8.8659 

o.59i3 

+8.2784 

+9.1906 

—9.3-711 

0.9216 

9.9587 

997 

9.o966 

9.4370 

+0.9253 

+9.o857 

+9.8981 

—  9.6o75 

O.92O7 

9.9589 

998 

9.o6i  i 

9.4022 

—0.2658 

—  9.0482 

—9.9945 

+9.6o5o 

0.9201 

9.959o 

999 

8.7681 

9.  I092 

+9.9012 

—  8.7i34 

—  o.ooSo 

+9.5632 

0.9201 

9.9590 

IOOO 

—  8.457i 

—8.  8012 

+0.5348 

+7-9°48 

—9.2074 

—  9.o63i 

+  0.9176 

—  9.9595 

440 


TABLE  XXX. 


CATALOGUE  OP  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.. 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.        s. 

s. 

IOOI 

55i9 

9  Ophiuchi,           w 

5 

16  23  16.10* 

+3.546 

Ill       8    25.8*+    8.l4 

IOO2 

6620 

10  Ophiuchi,           A 

4 

23    21.  16* 

3.027 

87  4i     o.7* 

8.27 

ioo3 

5523 

3o  Herculis,           g 

5 

23    43.05* 

1.969 

47  47    7.6* 

8.o9 

ioo4 

6626 

27  Herculis,            (3 

2* 

23  46.43 

2.676 

68   10  45.9 

8.17 

1006 

5532 

29  Herculis,           h 

4* 

26  35.  4i 

2.806 

78  ii     6.6 

8.08 

1006 
1007 

5536 

5538 

Triang.  Aust.,    7?1 
Scorpii, 

5 
5 

26     3.79 
26  3o.8i 

6.166 
3.932 

167  69  26.4 
124   56  29.5 

8.26 
8.09 

1008 

5539 

23  Scorpii,              r 

3* 

26  33.i3* 

+3.726 

117   53  58.o* 

7.94 

1009 

5545 

1  5  Draconis,           A 

4* 

28  18.10* 

—  o.i48 

20  54  26.9* 

7-79 

IOIO 

5547 

12  Ophiuchi, 

5 

28  28.89 

+3.i44 

91   69  69.1 

8.12 

IOII 

5548 

1  3  Ophiuchi,           £ 

H 

28  54.2i* 

3.299 

100    i5   3i.3* 

7.73 

1012 

5552 

35  Herculis,            a 

4 

29  16.10* 

1.932 

47    16     3.i* 

7.69 

ioi3 

5554 

Arae, 

4 

29  4i  .91 

5.263 

160  37   16.4 

7.70 

1014 

5578 

Triang.  Aust.,    a 

2 

32  5o.i3 

6.263 

168  44  34.6 

7.53 

ioi5 

5579 

Ophiuchi, 

5 

32  54.17* 

3.46i 

107   26  48.7* 

7.4i 

1016 

5596 

42  Herculis, 

5 

34  4o.64* 

i  .629 

4o  46  34.o* 

7.3i 

1017 

56o4 

4o  Herculis,            £ 

3 

35  37.99* 

2.266 

58     7  20.4* 

6.79 

1018 

56o9 

Arae,                    77 

4* 

36  5i.64 

5.i35 

i48  45  55.8 

7.i3 

1019 

56i7 

44  Herculis,            77 

3 

37  45.  3o* 

2.064 

5o  47  22.3* 

7.11 

IO20 

6621 

43  Herculis,            i 

5 

38  38.  o5 

2.876 

81     8  22.9 

6.93 

I  O2  I 

6628 

1  8  Draconis,           g 

5 

39  53.42* 

o.393 

26     7  34.5* 

6.92 

IO22 

5632 

26  Scorpii,              e 

3 

4o  27.43* 

3.875 

124     o  58.o* 

7.12 

1023 

5637 

20  Ophiuchi, 

5 

4i   32.  39* 

3.3i2 

100  3o  46.6* 

6.81 

1024 

5638 

Scorpii,               /zl 

3 

4i  43.  i5 

4.047 

127  47     4.i 

6.85 

IO25 

564o 

Scorpii,              /za 

4 

42  11.07 

4.o44 

127  45   22.0 

6.  79 

1026 

5643 

Draconis, 

5 

42  26.93 

i  .  126 

32   56  57.5 

6.66 

IO27 

5648 

47  Herculis,            k 

5 

43     2.45 

2.909 

82  29   19.7 

6.69 

1028 

565i 

Scorpii,               C1 

i* 

43  26.40 

4.202 

i32     6  24.8 

6.72 

1029 

566i 

Scorpii,               £2 

3 

44    2.33 

4.197 

i32     5  56.3 

6  .90 

io3o 

5666 

5o  Herculis, 

5 

44  47.8i 

2.338 

69  56     4.4 

6.46 

io3i 

5667 

62  Herculis, 

5 

44  60.79* 

1.745 

43  45   10.  i* 

6.53 

io32 

5683 

Arae,                    f 

31 

46  i3.56 

4.926 

i45  44  45.2 

6.4i 

io33 

5688 

23  Ophiuchi, 

5 

46  34.93 

3.203 

95  54  i4.5 

6.4* 

io34 

5692 

26  Ophiuchi,  '          i 

4 

46  54.8i* 

2.836 

79.  35     1.2* 

6.33 

io35 

5693 

53  Herculis, 

5 

47  i6.85* 

2.269 

58     2  49.1* 

6.26 

io36 

5697 

Arae,                    e1 

4 

47  38.  81 

4.762 

142  55  23.3 

6.28 

1037 

57o8 

27  Ophiuchi,           K 

4 

5o  34.21* 

2.838 

80  23  i5.8* 

6.96 

io38 

57i3 

Arae,                    e2 

5 

5i   10.97 

4.766 

i43     o  14.9 

6.99 

io3g 

573i 

58  Herculis,            e 

3 

54  33.10* 

2.294 

58  5o  58.2* 

5.58 

io4o 

5735 

Scorpii, 

5 

54  57.65* 

3.926 

123  54  23.o* 

6.67 

io4i 

574o 

I9  Draconis,           h1 

5 

55  12.74* 

o.3o3 

24  38     8.6* 

5.58 

IO42 

5747 

69  Herculis,            d 

5 

56     4.07* 

2.  211 

56  12  4i.i* 

5.49 

1043 

5765 

60  Herculis, 

5 

58  25.43 

2.781 

77     2  55.6 

5.3o 

1044 

6778 

Scorpii,              77 

31 

17     i  24.62 

+4.271 

i33     2     4.8 

5.4o 

io45 

578o 

22  Ursae  Minoris,   e 

4 

i  31.26* 

—  6.522 

7  43  27.9* 

5.o6 

io46 

6781 

35  Ophiuchi,           77 

2! 

i   46.  73* 

+3.435 

106  32     2.3* 

4.92 

io47 

5785 

21  Draconis,           /tz1 

4 

2   13.66* 

1.236 

35  i9  60.7* 

4-97 

io48 

5788 

Herculis, 

5 

2    42.98 

2.  123 

53  62     o.o 

4.98 

1049 

6802 

37  Ophiuchi, 

5 

5  23.65 

2.825 

79  i  3  43.2 

4.74 

1060 

58o8 

36  Ophiuchi,           A1 

41 

17     6     7.66* 

+3.683 

Il6    22    37.3* 

+  5.8i 

TABLE   XXX. 


441 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

V               c' 

d' 

IOOI 

—8.4667 

—  8.8i43 

-fo.5493 

+8.0237 

-8.8797 

—9.1696 

+0.914-7 

—  9.96oi 

IO02 

8.4363 

8.7845 

0.4802 

—7.0429 

—  9.6726 

+8.218-7 

0.9143 

9  .96o2 

ioo3 

8.5648 

8.9148 

0.2929 

—8.3921 

—9.9708 

+9.4378 

0.912-7 

9.9605 

100^ 

8.4665 

8.8167 

o.4i  19 

—  8.o367 

—  9.8706 

+9.i8o5 

0.9125 

9.9605 

ioo5 

8.4357 

8.7953 

0.449^! 

-7.7469 

-9.7870 

+8.9i37 

o.9o47 

9.9621 

1006 

8.85o6 

9.2126 

o.7856 

+8.8i77 

+9.8374 

—  9.5676 

0.9026 

9.9625 

1007 

8.5o87 

8.873i 

0.5942 

+8.2667 

+9.23o7 

—9.3564 

0.900-7 

9.9628 

1008 

8.4759 

8.84o4 

-|-o.57o6 

+8.i46o 

+8.6637 

—9.2682 

o.9oo5 

9.9629 

1009 

8.8619 

9.2357 

—9.1841 

—8.8324 

—0.0066 

+9.56io 

0.8928 

9.9643 

IOIO 

8.4i39 

8.7886 

+0.4933 

+6.9567 

-9.6o37 

—8.1325 

0.8919 

9.9644 

IOII 

8.4187 

8.7957 

o.5i77 

+  7-6694 

—9.4196 

—8.8385 

0.8900 

9.9648 

IOI2 

8.5442 

8.9231 

0.2856 

-8.3759 

—  9-9754 

+9.4179 

0.8884 

9.965o 

ioi3 

8.7174 

9.0986 

O.72l3 

+8.6576 

+  9.772I 

—9.5244 

0.886^ 

9.9654 

ioi4 

8.8342 

9.2324 

0.7967 

+8.8o36 

+9.85oi 

—9.539o 

o.87i8 

9.9678 

ioi5 

8.4i37 

8.8122 

0.5392 

+  7.8905 

—  9.i323 

—9.0462 

o.87i5 

9-9679 

1016 

8.5697 

8.9781 

O.2Il3 

—8.4490 

—  9.995i 

+9.44oi 

o.863o 

9.9692 

1017 

8.45io 

8.8648 

0.3607 

—  8.i738 

—  9.935i 

+9-2789 

o.8584 

9.9699 

1018 

8.6592 

9-°799 

o.  7103 

+8.59i2 

+9.7592 

—9.4820 

0.8523 

9.9-708 

1019 

8.48o3 

8.9061 

o.3n6 

—8.2811 

—  9  ,9676 

+9-3464 

o.8478 

9.9714 

IO20 

8.3702 

8.8012 

0.4587 

-7.5578 

-9.7583 

+8.7287 

0.8433 

9.9-720 

1021 

8.73o5 

9.1688 

9.5935 

—  8.6874 

—  o.oi49 

+9.4916 

0.8368 

9-9729 

1022 

8.437i 

8.8787 

0.5932 

+8.i848 

+  9.22I2 

—9.2-794 

o.8339 

9.9733 

1023 

8.3572 

8.8o53 

o.Sigi 

+7.6i84 

—  9.4o62 

-8.787i 

0.8282 

9.9740 

IO24 

8.45n 

8.9003 

0.6072 

+8.2384 

+  9.3698 

—  9«3l22 

0.82-72 

9.9742 

1025 

8.4485 

8.9004 

0.6071 

+8.2354 

+  9  .3698 

—  9.3095 

0.8247 

9.9745 

1026 

8.6095 

9.o63i 

o.oSio 

—8.5333 

—  O.OIlS 

+9.4449 

0.8233 

9-9747 

1027 

8.3455 

8.8o27 

o.463o 

—7.4619 

—  9<7432 

+8.6342 

0.8201 

9.9760 

1028 

8.4694 

8.9289 

0.6245 

+8.2958 

+9.4933 

—  9.3422 

0.8180 

9.9753 

1029 

8.4659 

8.9292 

0.6246 

+8.2923 

+9.4940 

—9.3388 

o.8i47 

9-9757 

io3o 

8.3949 

8.8629 

0.3688 

—  8.0947 

-9.0294 

+9.2081 

o.8io5 

9.9762 

io3i 

8.4921 

8.9603 

0.2427 

—  8.35o8 

—9.9930 

+9.3667 

0.8102 

9.9762 

1032 

8.5737 

9.o5o6 

0.6931 

+8.4910 

+  9.7322 

—9.4175 

0.8024 

9-9771 

io33 

8.3244 

8.8o36 

o.5o54 

+7.3367 

—9.5239 

—  8.5io4 

o.8oo4 

9-9773 

io34 

8.3374 

8.8o87 

0.4529 

-7.5846 

—  9-7771 

+8.7535 

o.7985 

9-9775 

io35 

8.3894 

8.873o 

o.3576 

—  8.u3i 

—9.9400 

+  9.2I78 

o.7964 

9-9778 

io36 

8.5357 

9.021-7 

o.6769 

+8.4376 

+9.6962 

—9.3940 

0.7943 

9.978o 

io37 

8.3o47 

8.8098 

0.4555 

—  7.5274 

—9.  -7689 

+8.6974 

o.7769 

9-9798 

io38 

8.5i55 

9.0246 

o.678i 

+8.4i78 

+9.  7oo5 

—  9.3733 

o.7732 

9.98oi 

1039 

8.34i3 

8.8736 

o.36o8 

—  8.o55i 

—9.9384 

+9.I635 

O.7520 

9.9821 

io4o 

8.3520 

8.887x 

o.5948 

+8.o985 

+9.2438 

—  9.1937 

o.7494 

9.9823 

io4i 

8.6495 

9.1863 

9.  433d 

—8.6080 

—  0.0228 

+9.4o4i 

o.7478 

9.9824 

IO42 

8.344i 

8.8871 

0.3445 

—8.0893 

—  '9  .9520 

+9.i85i 

0.742I 

9.9829 

io43 

8.259i 

8.8192 

o.443i 

—  7.6096 

—  9.8o55 

+8.7745 

o.  -7262 

9.984i 

io44 

8.3629 

8.9457 

+o.63i3 

+8.1970 

+9.5354 

—  9.237o 

o.7o5i 

9.9857 

io45 

9.o976 

9.6812 

—  o.8io7 

—9.0937 

—  o.oo74 

+9.3982 

o.7o44 

9.9857 

io46 

8.24o3 

8.8259 

+0.5353 

+  7.6681 

—  9.2011 

—8.8280 

O.7O24 

9.9858 

1047 

8.4587 

9.0478 

0.0949 

—  8.37o2 

—  o.oi75 

+9.3o85 

o.699i 

9.986i 

io48 

8.3ioo 

S.goSo 

o.3273 

—8.0806 

—9.964-7 

+9.1639 

o.6955 

9.9863 

1049 

8.2o46 

8.8192 

o.45o8 

—7.4762 

—  9>7838 

+8.6445 

0.6-762 

9.9876. 

io5o 

—8.2388 

—  8.8595 

+o.57oo 

+  7.8864 

+8.6365 

—  9.0148 

+o.6694 

—9-9879 

442 


TABLE  XXX. 


CATALOGUE  OP  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.. 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.        s. 

3. 

o           /            Ti 

io5i 

58io 

Apodis,              £ 

4 

17     6  22.  89 

-f-6.258 

157  36  i9.4 

+  4.69 

1062 

6821 

64  Herculis,            a 

9* 

7  48.53* 

2.734 

75  26     4.9* 

4.48 

io53 

5823 

22  Draconis,           f 

3 

8    21.  0.2* 

o.  i59 

24     6     1.6* 

4.47 

io54 

5828 

65  Herculis,            6 

4 

8    52.  29 

2.459 

64  58  49.8 

4.59 

io55 

583o 

4  1  Ophiuchi, 

4* 

8  55.i7 

3.082 

9o  16  i6.5 

4-49 

io56 

5834 

67  Herculis,            TT 

3* 

9  49.5i 

2.088 

53     i     5.5 

4.32 

1067 

5842 

68  Herculis,           u 

4 

ii  47.24 

2.212 

56  44     5.i 

4.i7 

1068 

5844 

4o  Ophiuchi,           £ 

4* 

12        I  .  o4* 

3.592 

no  56  47.  9* 

4.36 

1069 

5845 

53  Serpentis,          v 

4* 

12  23.58* 

3.371 

102    4l     22.4* 

4.  ii 

1060 

5847 

69  Herculis,            e 

41 

12  3o.o3 

2.067 

52    32    52.1 

4.02 

1061 

585o 

Arae,                    y 

3 

12  46.  98 

5.O27 

i  46   i3  43.8 

4.10 

1062 

585i 

£2  Ophiuchi,           6 

w 

12    48.07* 

3.680 

n4  5o  39.o* 

4.i5 

io63 

5852 

Arae,                   /3 

3 

12  5o.58 

4.966 

i45  22  5o.8 

4.i9 

1064 

5859 

Arae,                    KI 

5 

i4  18.14 

4.6i4 

l4o    29    25.  0, 

4.o4 

1066 

5876 

44  Ophiuchi,           b 

5 

17  i2.69* 

3.66o 

n4     i  54.4* 

3.8o 

1066 

5877 

Arae,                    d 

4 

17  34.  5i 

5.39o 

i5o  33     0.2 

3.8o 

1067 
1068 

588i 
5886 

45  Ophiuchi,           d 
75  Herculis,           p 

4 
4 

17  46.77 
18  3o.68 

3.8i9 
2.073 

ii9  43   32.  9 

52    42    45.7 

3.87 
3.59 

1069 

5893 

49  Ophiuchi,           a 

4^ 

19     4.4?* 

2-977 

85  43  3o.2* 

3.54 

1070 

5899 

Arae,                    a 

3 

20  i5.4a 

4.625 

i39  45     o.5 

3.56 

1071 

5901 

34  Scorpii,               v 

3* 

20  34.  3i 

4.071 

127  10     5.6* 

3.5i 

1072 

59°7 

5  1   Ophiuchi,           c2 

5 

22     l6.0I* 

3.66o 

ii3  5o  28.2* 

3.3i 

1078 

59i5 

35   Scorpii,              A 

3 

23    25.  90 

4.072 

126  59   16.1* 

3.i4 

1074 

5922 

76  Herculis,            a 

ft 

24  4o.6i* 

2.421 

63  46   22.0* 

3.oi 

1076 

5935 

Scorpii,              6 

3 

26  32.  80 

4.3oi 

i32  53  45.6 

3.o6 

1076 

5937 

23  Draconis,           P 

2* 

27     2.72* 

1.353 

37  35     8.2* 

2.86 

1077 

594i 

55  Ophiuchi,           a 

2 

27  58.35* 

2.781 

77   i9   36.o* 

2.98 

1078 

5949 

55  Serpentis,          § 

5 

28  59.96* 

3.433 

io5   17   56.i* 

2.75 

1079 

595o 

24  Draconis,           v1 

5 

29  i3.33* 

1.170. 

34  42  4i.i* 

2.66 

1080 

595i 

25  Draconis,           va 

5 

29  i8.79* 

I-I79 

34  43  23.7* 

2.67 

1081 

5953 

57  Ophiuchi,           p. 

5 

29    41.67 

3.259 

98     i   18.9 

2.63 

1082 

5959 

Octantis,             a 

6 

3o     4.o9 

107.504 

i79   16  21  .9 

2.61 

io83 

5963 

Pavonis,             ?/ 

41 

3i     1.62 

5.863 

i  54  38  4o.o 

2.71 

io84 

597o 

Scorpii,               K 

3 

32     7.07 

+4.146 

128   56  47.4 

2.5l 

io85 

5972 

27  Draconis,          f 

5 

32    34.21* 

—  O.256 

21   46   11.  5* 

2.28 

1086 

5976 

56  Serpentis,          o 

41 

32  59.i7* 

+  3.369 

IO2    47    24.5* 

2.38 

1087 

5987 

58  Ophiuchi, 

5 

34  26.58* 

3.594 

in    36   i6.5* 

2.16 

1088 

599o 

85  Herculis,           t 

4 

35  i4.n* 

1.713 

43  54  4i.2* 

2.l5 

1089 

5996 

69  Ophiuchi,           (3 

3 

36     3.68* 

2.964 

85  21   56.7* 

1.92 

1090 

6oo4 

Scorpii,              i1 

H 

37     6.o5 

+4-200 

i3o     3  47.3 

2.14 

1091 

6006 

28  Draconis,           « 

4 

37  5o.o3* 

—0.368 

21     10    23.7* 

.66 

1092 

6008 

3  Sagittarii, 

5 

38     7.11* 

+  3.767 

117  46     4-8* 

.9i 

1098 

6018 

Scorpii, 

4 

39  39.o9 

4.o83 

126    59    29.2* 

.85 

1094 
1096 

6020 
6021 

62  Ophiuchi,           7 
86  Herculis,           p 

4 
4 

4O    22.23* 

4o  35.46 

3.oo4 
+2.344 

87   i3  55.2* 
62   ii    16.4 

-79 
.42 

1096 

6o47 

3  1  Draconis,           V1 

ft 

44  36.63 

—  i.o96 

17  46  45.  9* 

.60 

1097 

6074 

Sagittarii, 

5 

49  27.22* 

+3.852 

120  i3  55.4* 

.00 

1098 

6077 

4  Sagittarii, 

5 

5o  38.o9* 

3.66i 

u3  47  48.3* 

o.83 

1099 

6078 

64  Ophiuchi,           v 

4 

5o  46.22*; 

3.3o4 

99  45     1.7* 

o.9i 

1  1  00 

6o79 

32  Draconis,            £ 

M 

7  5o  56.33* 

+  i.o36 

33     6     8.0* 

+  o.73 

TABLE    XXX. 


443 


CATALOGUE   OF    1500   STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a! 

V 

c' 

d' 

io5i 

—8.6082 

—  9.  23io 

+  0.7943 

+8.574i 

+9.  8658 

—9.33n 

+0.6674 

—9.9880 

io52 

8.1917 

8.8267 

0.4365 

—  -7.5923 

—9.8228 

+  8.7542 

o.6559 

9.9886 

io53 

8.6620 

9.2018 

9.1967 

—8.5224 

—  0.0282 

+9.3095 

o.65i3 

9.9889 

io54 

8.2116 

8.8558 

o.SgiS 

-7.8378 

—  9.9o6i 

+8.9711 

0.6471 

9.989i 

io55 

8.i684 

8.8i3o 

o.488i 

+  5.8439 

—  9.6329 

—  7.0199 

0.6467 

9.989i 

io56 

8.2583 

8.9110 

o.3i97 

—  8.o376 

—9.9703 

+  9.11.61 

0.6391 

9.9895 

io57 

8.2214 

8.8920 

o.345o 

—  -7.9606 

-9.9539 

+9.o59o 

0.6220 

9.99o3 

1068 

8.1713 

8.844o 

0.5528 

+7.7246 

—  8.745i 

—8.8710 

0.6199 

9.9904 

1069 

8.1490 

8.8252 

O.527O 

+7.4908 

—9.3191 

-8.656i 

0.6166 

9.99o6 

1060 

8.2376 

8.9148 

o.3i56 

—8.o2i5 

—9.9729 

+9.0974 

o.6i56 

9.99o6 

1061 

8.3898 

9.0697 

o.7oi4 

+8.3o95 

+9.7578 

—  9.23o6 

o.6i3i 

9.990-7 

1062 

8.1768 

8.8568 

0.5655 

+  7.8oo2 

+8.243o 

—  8.934i 

0.6129 

9«99°7 

io63 

8.  3798 

9  .0602 

o.696o 

+8.295i 

+9.7475 

—  9.2257 

o.6i25 

9>99°7 

io64 

8.3i7i 

9.0116 

0.6684 

+8.2045 

+9.68i3 

—9.1842 

0.5991 

9.99i3 

io65 

8.i3i9 

8.8557 

o.563i 

+  7.74i7 

+  7-5798 

—8.  8784 

0.5708 

9.9924 

1066 

8.3972 

9.1248 

O.7322 

+8.337i 

+  9.80-79 

—9.2049 

0.5672 

9.9925 

1067 

8.i48i 

8.8778 

O.5822 

+  7-8434 

+  9.o438 

—  8.9582 

o.565i 

9.9926 

1068 

8.1786 

8.9161 

o.3i58 

—  7.96o9 

—  9«9737 

+9-o377 

0.5576 

9.9928 

1069 

8.0746 

8.8182 

o.473i 

—  6.947o 

—  9.7o43 

+8.1219 

o.55i7 

9'993° 

1070 

8.25o5 

9.0070 

o.  665.2 

+8.I332 

+9.6734 

—9.1196 

0.5391 

9.9934 

1071 

8.i56o 

8.9161 

o.6o95 

+7.9372 

+9.397I 

—  9.oi47 

o.5357 

9.9935 

1072 

8.0772 

8.8567 

o.5627 

+7.6838 

+  7.2553 

—8.8211 

o.5i68 

9.994i 

io73 

8.1226 

8.9159 

0.6091 

+7-9OI9 

+9.3934 

—8.  98o4 

o.5o33 

9.9945 

1074 

8.0573 

8.8659 

0.3838 

—  7.7026 

-9.9170 

+8.83i5 

o.4884 

9.9948 

1076 

8.1218 

8.9544 

0.6334 

+  7.9547 

+9.55o8 

-8.9957 

0.4649 

9.9954 

1076 

8.1948 

9.o34i 

o.i3o8 

—  8.o938 

—  0.0208 

+9.o552 

o.4584 

9.9955 

1077 

7-9785 

8.83o4 

0.4429 

—  7.3i98 

—9.8067 

+8.4852 

o.446i 

9.9958 

1078 

7.9694 

8.8356 

o.5357 

+7.3908 

—  9.  1965 

—  8.55i2 

0.4321 

9.996o 

1079 

8.i952 

9.o645 

o.o637 

—  8.  noo 

—  0.0260 

+9.04*6 

0.4289 

9.996i 

1080 

8.1938 

9.0644 

o.o64o 

—8.1086 

—  0.0260 

+9.o4o3 

0.4277 

9.996i 

1081 

7-9483 

8.8244 

0.5129 

+  7.0930 

-9.  4645 

—8.2648 

0.4223 

9.9962 

1082 

9.835i 

0.7167 

2.o3i4 

+9.835i 

+9-9938 

—9-II47 

O.4l69 

9.9963 

io83 

8.2929 

9.1888 

o.7686 

+8.2489 

+  9.85i7 

—  9.0567 

O.4O29 

9.9965 

io84 

8.0172 

8.9299 

+0.6173 

+7.8i55 

+9.457I 

—8.8824 

0.3863 

9.9968 

io85 

8.33i6 

9«25i6 

—  9»4o36 

—8.2995 

—  o.o346 

+9-0448 

o.3792 

9.9969 

1086 

7.9052 

8.83i8 

+0.5279 

+7.25o3 

—9.3o84 

—  8.4i55 

0.3-726 

9*997° 

1087 

7.9019 

8.8528 

o.5559 

+  7.4680 

-8.5752 

—8.6i25 

0.3486 

9.9973 

1088* 

8.oi56 

8.9803 

O.2279 

—7.8732 

—  o.oo74 

+8.8903 

o.335o 

9.9975 

1089 

7.8434 

8.823o 

0.4717 

—6.7508 

—  9.7ioo 

+  7.9254 

o.32o3 

9.99-76 

1090 

7.9390 

8.9379 

+  O.6222 

+  7.7476 

+9.4890 

—8.8076 

o.3on 

9.9978 

1091 

8.25io 

9  .2641 

—  9.  5623 

—8.2206 

—  o.o354 

+8.9545 

O.287O 

9.9980 

1092 

7.8562 

8.8751 

+0.5765 

+7.5245 

+8.9020 

—8.6475 

0.2814 

9.9980 

1093 

7.8693 

8.9i98 

0.6100 

+7-6487 

+9.4026 

—8.7272 

o.25oo 

9.9983 

109^ 

7.7566 

8.8228 

0.4780 

—  6.44o5 

—  9-6827 

+  7.6161 

0.2344 

9.9984 

1096 

7.8o45 

8.8757 

+o.3743 

-7.4734 

—9.9288 

+8.5962 

0.2295 

9.9984 

1096 

8.1669 

9.338i 

i—  0.0873 

—8.1446 

—  o.o345 

+8.8o55 

o.  1290 

9.9990 

1097 

7.55oo 

8.8869 

+0.5853 

+7.2520 

+9.io76 

—8.3646 

9.9649 

9-9995 

1098 

7.4737 

8.8621 

0.5634 

+7.0795 

+  7.7782 

—8.2170 

9.9134 

9«9996 

1099 

7.435i 

8.8299 

o.5i85 

+6.6639 

—9.4125 

-7.  8337 

9.  9o7i 

9-9997 

IIOO 

—  7.6836 

—  9.o863 

+O.O092 

—  7.6066 

—  o.o3io 

+8.5200 

+9.8992 

—9-9997 

444 


TABLE  XXX. 


CATALOGUE   OF    1500    STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.. 
Jan.  1,  1850. 

Annual 
Variat'n. 

k.       771.              S. 

s. 

0            '                // 

" 

IIOI 

6082 

91   Herculis,           6 

4 

17  5i     6.67 

4-  2.o55 

52  43  35.4* 

+o.73 

IIO2 

6o84 

92  Herculis,           f 

4 

5i  56.24 

2.332 

60  43  56.6 

0.73 

no3 

6o85 

67  Serpentis,          f 

5 

52  33.77 

3.169 

93  4o  3i  .0 

0.68 

no4 

6087 

94  Herculis,           v 

5 

52  45.83* 

2.296 

59  4?  42.3* 

o.63 

no5 

6089 

66  Ophiuchi, 

5 

52  So.og* 

2.971 

85  37    4.1 

0.62 

1106 

6091 

33  Draconis,           y 

2 

53     7.45* 

1.394 

38  29  29.4* 

o.63 

1107 

6092 

67   Ophiuchi, 

4 

53     8.35 

S.oio 

87     3  24.5 

o.63 

1108 

6094 

93  Herculis, 

5 

53  22.69* 

2.667 

73    l4    12.0* 

o.5i 

1109 

6100 

Pavonis,              TT 

5 

54     7.20 

5.7i5 

i53  39  52.2 

o.57 

IIIO 

6io4 

69  Ophiuchi,           T 

5 

54  55.01 

3.269 

98   10  27.9 

0.43 

mi 

6io5 

Arae,                   6 

4 

54  57.59 

4.678 

i4o    5  41.2 

0.52 

III2 

6107 

Sagittarii,          yl 

4 

55  26.54 

3.84o 

119  34  53.o 

o.48 

iii3 

6110 

96  Herculis, 

5 

55  58.44 

4-  2.564 

69     9  44.0 

o.3o 

in4 

6ii4 

35  Draconis, 

5 

56  10.  3o* 

—  2.702 

1  3     i   16.2* 

0.08 

iii5 

6ii5 

10  Sagittarii,          -f 

4 

56  10.68* 

4-   3.858 

I2O    25     12.5* 

o.57 

1116 

6i23 

70  Ophiuchi, 

41 

57  52.35* 

3.028 

87  27  38.  7* 

1.28 

1117 

6126 

Pavonis,             t 

41 

58  20.86 

5.539 

i5i   33  34.9 

o.i5 

1118 

6127 

Sagittarii, 

5 

58  34.97 

3.795 

118  28     2.9 

0.20 

1119 

6129 

Hereulis, 

5 

59  i3.82* 

1.562 

4i   32  25.7* 

0.07 

II2O 

6i4o 

TeJescopii,         e 

41 

18    •*>     5.90 

4.456 

i35  58  24.7 

+0.04 

II2I 

6i43 

72  Ophiuchi, 

4 

o  i4.4i* 

2.844 

80  27   i3.2* 

0.  10 

1122 

6i48 

Pavonis, 

5 

i  23.  3i 

5.639 

i53     5     6.9 

O.  I  I 

1123 

6i5o 

io3   Herculis,           o 

4 

i  4i.53* 

2.34i 

61    i5  16.8* 

0.17 

1124 

6168 

1  3   Sagittarii,          fi 

31 

4  47.57* 

3.587 

in     5  33.8* 

0.43 

1125 

6178 

io4  Herculis,            A 

5 

6  i5.65 

2.261 

58  37  43-7* 

0.62 

1126 

6186 

Sagittarii,          77 

4 

7  28.60 

4-  4.o56 

126  48     8.2* 

0.42 

1127 

6206 

4o  Draconis, 

5 

ii   i5.4o 

-  4-459 

10       I     33.4 

i  .04 

1128 

6208 

4  1    Draconis, 

5 

ii  21  .57 

—  4.466 

10     121.7 

i.o5 

1129 

6209 

19  Sagittarii,          6 

31 

ii  23.43* 

4-  3.842 

119  53   10.6* 

0.94 

ii3o 

6218 

Lyrae, 

5 

12    21.  4l* 

i  .915 

49     7  11.7* 

i.  08 

u3i 

6223 

io5  Herculis, 

5 

i3     o.36* 

2.471 

65  36  45.7* 

i.i4 

1  1  32 

6224 

36  Draconis, 

5 

i3     1.90* 

o.345 

25  39  ii.  i* 

i.i4 

ii33 

6229 

58   Serpentis,          rj 

4 

i3  33.i4* 

3.102 

92  56     o.3* 

o.54 

ii34 

6233 

20  Sagittarii,          e 

3 

i4  12.94* 

3.987 

124  27     i  .0* 

1.16 

ii35 

6235 

i   Lyrae,                 K 

4^ 

i4  36.38* 

2.103 

54       0       0.2* 

i.3i 

ii36 

6240 

Telescopii,         a 

4 

i5  50.92 

4.45i 

i36     2  42.3 

i.i5 

1  1  87 

6260 

Telescopii,         f 

41 

17  iS.oi 

4.597 

i3g     8  42.5 

i  .07 

ii38 

6253 

Pavonis,             v 

5 

17    21  .98 

5.644 

i52  21   55.6 

i.35 

1  1  39 

6255 

Draconis, 

5 

17  42.  5o* 

1.535 

4o  57     7.5* 

i.55 

n4o 

6263 

22  Sagittarii,          A 

4 

18  42.75* 

3.7o7 

n5  29  56.7* 

i.4i 

n4i 

6278 

Telescopii,         (51 

5 

20  38.4? 

4-445 

i36     o  34.3 

i.64 

1142 

6279 

Sagittarii, 

5 

20  38  96* 

4-  3.4i9 

io4  39  23.9* 

1.79 

ii43 

6281 

23  Ursae  Minoris,    6 

3 

20    43.54* 

—19.293 

3  24   10.0* 

i.83 

1144 

6282 

Telescopii,         (53 

5 

20  56.19 

4-  4-438 

i35   5i    i3.5 

i.73 

n45 

6289 

39  Draconis,           b 

5 

21    42.93* 

4-  0.876 

3i    17     5.6* 

i  .94 

n46 

6296 

Coronae  Aust.,  6 

5 

22  47-97 

4-  4.3oi 

l32    24    52.2 

1.86 

1  1  47 

6297 

43  Draconis,           $ 

5 

22    54.25 

—  o.848 

18  44  36.9 

2.01 

ii48 

63o2 

44  Draconis,           % 

41 

23  45.53 

—  1.073 

17  19  59.1 

i.73 

n49 

63i5 

Pavonis,       -     f 

4 

25  29.29 

4-  7-°45 

161  32  44.6 

2.07 

ii5o 

635o 

Draconis,                    5 

18  3o  32.  o5 

4-  i.36o 

37  45  5i.o 

—2.66 

TABLE   XXX. 


445 


CATALOGUE   OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b         |         c 

d 

a! 

b' 

c' 

d' 

IIOI 

—  7.6117 

—8.9228 

4-o.3i26 

—7.2939 

—  9.978o 

+8.37o8 

+9.89o8 

—9-9997 

IIO2 

7.4294 

8.883o 

0.3658 

—7.1186 

—9.9381 

+8.2354 

9.  8484 

9-9997 

no3 

7.3359 

8.8246 

0.4992 

+6.1428 

—  9.567o 

—  7.3i8o 

9-8i33 

9-9998 

1104 

7.3864 

8.8871 

o.36o3 

—7.0881 

-9.9434 

+8.2008 

9.8014 

9-9998 

no5 

7-3i99 

8.8260 

0.4726 

—  6.2o3i 

—  9.7068 

+  7.3779 

9-7970 

9-9998 

1106 

7.6068 

9.0296 

o.i43o 

—  7«4oo4 

—0.0222 

+8.37o6 

9.7792 

9-9998 

1107 

7.3oo6 

8.8243 

0.4774 

—  6.0II2 

—9.6866 

+7.i867 

9.7784 

9-9998 

1108 

7.3o35 

8.8426 

0.4262 

—  6.7636 

-9.8475 

+  •7.9208 

9.7630 

9-9998 

1  109 

7.6860 

9.1768 

0.7612 

+  7.5385 

+9-8472 

—  8.36i6 

9.7114 

9-9999 

IIIO 

7.1743 

8.8282 

o.5i36 

+6.3271 

—9.4686 

-7.4988 

9.6482 

9-9999 

mi 

7.3689 

9.0166 

0.6692 

+7.2438 

+9.6889 

—  8.227I 

9-6444 

9-9999 

III2 

7.i83i 

8.8845 

o.583i 

+6.8766 

+9.0662 

—  7.9920 

9.6008 

9-9999 

ui3 

7.0979 

8.8532 

+o.4o86 

—  6.6490 

—9.8816 

+  7.7967 

9.6468 

9-9999 

iii4 

7.6946 

9.4711 

—  o.433o 

—7.6833 

—  o.o3o2 

+  8.2122 

9.6267 

9-9999 

i.iij5 

7.  1106 

8.8882 

+0.6861 

+6.8149 

+9.1209 

—  7.9267 

9.6245 

9-9999 

1116 

6.7914 

8.8243 

0.4788 

—6.4379 

-9.6793 

+6.6i36 

9.2693 

o.oooo 

1117 

7.oo4o 

9.1461 

0.7434 

+6.948i 

+9.8278 

—  7.8020 

9.1601 

o.oooo 

1  1  1.8 

6.6710 

8.8799 

0.6793 

+6.3492 

+8.9786 

-7.4693 

9.o933 

o.oooo 

1x19 

—  6.53ii 

9.0023 

0.1936 

—  6.4o53 

—  0.0166 

+  7.4o3o 

+8.83io 

o.oooo 

H20 

+5.6169 

8.9819 

0.6487 

—5.4736 

+9.6206 

+6.4917 

—  7.9372 

o.oooo 

I  121. 

5  .,848  o 

8.83oo 

O.4542 

+6.0677 

—9.7737 

-6.2377 

8.3202 

o.oooo 

1122 

6.  .9606 

9.1681 

0.7662 

—  6.9008 

+9.8422 

+  7.7327 

9.0847 

o.oooo 

1  1  23 

6.7499 

8.  8810 

0.3687 

+6.43i9 

—9.9362 

—  7.55o9 

9.1711 

o.oooo 

1124 

7.1746 

8.8539 

0.5546 

—  6.7307 

—8.6642 

+  7.8767 

9.6227 

9-9999 

1126 

7.3290 

8.8924 

0.3533 

+7.o455 

—9.9498 

—  8.i529 

9.7386 

9-9998 

1126 

7.4339 

8.9202 

+0.6096 

—  7.2113 

+9.4000 

+8.2909 

9.8i57 

9.9998 

1127 

8.2742 

9.6826 

—  o.65i5 

+8.2676 

—  0.0260 

—8.6844 

9.9933 

9.9996 

1128 

8.2783 

9.6827 

—  0.6618 

+8.2716 

—  0.0249 

—8.6883 

9-9972 

9.9996 

1129 

7.6821 

8.8854 

+o.584i 

—7.2796 

+9.0846 

+8.3937 

9.9984 

9.9996 

ii3o 

7.6772 

8.9447 

0.2822 

+  7-493i 

—9.9919 

-8.5477 

o.o34o 

9.9994 

ii3i 

7.6i83 

8.8638 

0.3919 

+7.2341 

-9.9071 

—  8.3696 

o.o56o 

9.9993 

1132 

7-9421 

9.1868 

9-4642 

+7.8970 

—  o.o366 

—  8.7o95 

0.0668 

9.9993 

ii33 

7.6960 

8.8237 

0.4968 

—  6.3o5o 

—9.6824 

+  7.48o5 

o.o737 

9-9992 

ii34 

7.7000 

8.9068 

0.6006 

-7.4626 

+9.3i53 

+8.5449 

0.0946 

9-9992 

ii35 

7.7201 

8.9161 

0.3226 

+7.4893 

—  9.9721 

—  8.5733 

o.io63 

9-9991 

ii36 

7.8220 

8.9816 

o.6488 

—7.6792 

+9.6201 

+8.6967 

o.i4i7 

9-999° 

ii37 

7.8844 

9.0070 

o.6639 

-7.7632 

+9.6724 

+8.7549 

o.i784 

9.9988 

u38 

8.0367 

9.i563 

0.7496 

-7.984i 

+9.8339 

+8.8266 

o.i8i3 

9.9988 

1139 

7.8949 

9.0061 

o.  1860 

+7.773o 

—  0.0160 

—  8.7656 

o.  i897 

9-9987 

n4o 

7-7799 

8.8670 

0.6689 

-7.4i39 

+8.6729 

+8.5455 

O.2l37 

9.9986 

n4i 

7.936-2 

8.9804 

o.6484 

—  7.7932 

+9.6182 

+8.8wo 

0.2662 

9.9982 

1X42 

7.7924 

8.8365 

+o.5339 

-7.i955 

—9.2248 

+8.3573 

0.2563 

9.9982 

ii43 

9  .0062 

o..o487 

—  1.2861 

+9.0064 

—0.0086 

—8.  9660 

0.2680 

9.9982 

n44 

7.9411 

8.9792 

+0.6476 

—7.7970 

+9.6i52 

+8.8160 

0.2623 

9.9982 

n45 

8.o845 

9.1065 

+9.9446 

+8.oi63 

—  o.o3i9 

—  8.9078 

o.2783 

9.9981 

ii46 

7.9627 

8.9535 

+0.6321 

—  7.78i7 

+9.545i 

+8.8260 

o.2993 

9-9979 

n47 

8.3i6o 

9-3:48 

—9.9293 

+8.2923 

—  o.o34i 

—  8.9753 

O.3OI2 

9-9978 

n48 

8.3647 

9.3476 

—  0.0767 

+8.3445 

—  0.0329 

—8.9947 

o.3i7i 

9-9977 

n49 

8.3687 

9.3208 

+0.8483 

—8.3458 

+9.9060 

+9.0223 

o.3475 

9-9973 

n5o 

+8.1602 

—  9.o33o 

+o.i335 

+  8.0681 

—  0.0209 

—  9.O2I2 

-0.4266 

—  9.9961 

446 


TABLE  XXX, 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

ft.     m.        s. 

s. 

O                 '                 '/ 

ii5i 

6352 

Pavonis, 

5 

18  3o  43.39 

+5.937 

i55     o     9.9 

—    2.70 

1162 

6355 

3  Lyras,                  a 

I 

3i   5i.56* 

2.O32 

5  i   21   10.9* 

3.o6 

n53 

636o 

Pavonis,             0 

5 

33  61.62 

5.888 

i55   i3  3i.3 

2.8o 

n54 

636i 

2  Aquilas, 

5 

34     3.69 

3.289 

99   n  27.5 

2.97 

n55 

637i 

27  Sagittarii,           0 

4* 

36  16.97* 

3.758 

117       8    22.3* 

3.i3 

u56 

6383 

Pavonis,              /I 

5 

38  19.00 

6.620 

l52    21        1.9 

3.33 

1167 

6387 

no  Herculis, 

5 

39  12.47 

2.585 

69  35  36.3 

3.o6 

n58 

6390 

4  Lyras,                  £* 

5 

Sg    22.12* 

1.986 

5o  29     2.1* 

3.5o 

1169 

639i 

5  Lyras,                  e2 

5 

39    24-46* 

1.988 

5o  32   28.1* 

3.5i 

1160 

6392 

6  Lyras,                  £ 

5 

39  36.35 

2.064 

52     32     52.4 

3.52 

1161 

6395 

46  Draconis,           c 

5 

39  43.  4i* 

i  .  160 

34  36   39.o* 

3.46 

1162 

64o5 

Pavonis,             K 

5 

4i  28.09 

6.269 

167  24  48.9 

3.53 

u63 

6419 

Draconis, 

5 

43  2i.55 

1.339 

37   10  29.9 

3.  77 

1164 

6429 

10  Lyras,                  j3 

3 

44  32.  5o* 

2.2l5 

56  48  3o.4* 

3.85 

n65 

6434 

32  Sagittarii,           v1 

5 

45    6.65* 

3.627 

112  55  26.9* 

3.93 

1166 

644o 

34  Sagittarii,          a 

3 

45  57.78* 

3.729 

116  28  38.8* 

3.92 

1167 

644i 

35  Sagittarii,          v* 

5 

46    2.95* 

3.632 

112  5i     9.8* 

4.02 

1168 

6452 

Draconis, 

5 

48  i  3.  02 

1.349 

37   i3   ii.  i 

4-i9 

1169 

6453 

n3  Herculis, 

5 

48  25.24 

2.535 

67  32  29.0* 

4.27 

1170 

646o 

63  Serpentis,          6l 

4* 

48  45.58* 

2.982 

85  69   14.1* 

4-34 

1171 

646i 

37  Sagittarii,           ? 

4 

48  46.64* 

3.583 

in    17  55.  o* 

4.24 

1172 

6462 

Serpentis, 

5 

48  47.04* 

2.9,82 

85  69  19.7* 

4.36 

n73 

6463 

47  Draconis,            o 

5 

48  58.86* 

0.886 

3o  47   37.8* 

4.25 

"74 

6466 

12  Lyras,                  <52 

5 

49  15.63 

4-2.  o99 

53   17   i8.3 

4.32 

1176 

6469 

Draconis, 

5 

49  28.82 

—  i.449 

16     5  22.1 

4.43 

1176 

6475 

1  3  Lyras, 

5 

5o  46.i6* 

+  1.825 

46  i4  56.7* 

4.4i 

1177 

6478 

5o  Draconis, 

5 

5r   10.70 

—  i  .9oi 

i4  44  49.7 

4-46 

1178 

6487 

1  3  Aquilas,               c 

31 

62  48.91* 

+2.723 

75     7  53.9* 

4-48 

1179 

6489 

38  Sagittarii,           £ 

H 

53     3.77* 

3.828 

120       5    20.3* 

4.57 

1180 

6491 

1  4  Lyras,                 y 

3 

53  20.  01 

2.244 

57  3o  45.3 

4.65 

1181 

64g6 

48  Draconis, 

5 

54  12.66* 

1.018 

32   22  67.5* 

4.63 

1182 

65o7 

39  Sagittarii,           o 

4* 

55  4i.43* 

4-3.600 

III     67    21.  O* 

4.8o 

u83 

65io 

52  Draconis,           v 

5 

56  i3.oi* 

—  o.  708 

18  54  16.0* 

4.92 

n84 

65n 

Coronas  Aust.,    y 

5 

56  16.72 

4-4.072 

127    l6    22.1 

4.5o 

n85 

652i 

4o  Sagittarii,           T 

4 

57  34.27* 

3.755 

117  53     2.7* 

4.76 

1186 

6523 

Coronas  Aust.,   <5 

5 

57  53.97 

4.i84 

i3o  43  23.8 

4.87 

1187 

6526 

1  6  Aquilas,               A 

3 

58  17.35 

3.187 

96     6   10.8 

4.98 

1188 

6528 

17  Aquilas,               £ 

3 

58  30.90* 

2.755 

76  21    19.9* 

5.01 

1189 

6535 

Coronas  Aust.,   a 

4* 

5g  15.91 

4.o98 

128     7  54.4 

4-97 

1190 

654i 

Coronas  Aust,  ft 

5 

59  42.46 

4.i43 

129  34  23.6 

5.  02 

1191 

6548 

4  1   Sagittarii,           TC 

41 

19     o  5o.33* 

3.675 

in    1  5  24.4* 

6.27 

1192 

6564 

20  Aquilas, 

5 

4  32.  4o* 

3.260 

98   n     7.6* 

5.59 

n93 

6575 

42  Sagittarii,           $ 

5 

6  20.44 

3.687 

116  3o  33.8 

5.  74 

1194 

658i 

20  Lyras,                 77 

5 

8  39.21 

2.042 

5i     6  32.7 

6.97 

1196 

6583 

53  Draconis, 

5 

8  50.09* 

i.i36 

33  23  4i.3* 

6.98 

1196 

6584 

43  Sagittarii,          d 

5 

8  61.29* 

3.619 

109   12   53.2* 

6.98 

1197 

6589 

i  Vulpeculas, 

5 

9  46.  18 

2.58i 

68  62   11.9 

6.o7 

1198 

6595 

25  Aquilas,               w 

5 

10  46.70 

2.818 

78  4o   i  3.  i 

6.16 

1199 

6599 

21   Lyras,                  d 

5 

ii     9.66* 

2.080 

52     7  5o.o* 

6.i3 

I2OO 

6601 

54  Draconis, 

5 

19  ii   i4«32* 

+  1-079 

32  33-    8.7* 

—  6.06 

TABLE   XXX. 


447 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a'         \         b'                  c'                  d' 

n5i 

4-8.324o 

—9.1941 

+0.7718 

—  8.28i3 

+9.8542 

+9.o833 

—  0.4282 

—9.9961 

1162 

8.0729 

8.9270 

o.3o36 

+7.8685 

—  9.9811 

-8.9372 

0.4439 

9.9968 

n53 

8.3696 

9.  1969 

o.7735 

—8.3276 

+9.855o 

+9.1260 

o.47o 

9.9962 

u54 

8  .  oooo 

8.8247 

o.5i65 

—  7.2034 

—9.4320 

+8.3739 

0.4727 

9.9962 

n55 

8.0728 

8.8691 

o.5737 

—  7.73i4 

+8.8082 

+8.8569 

o.5ooo 

9.9945 

n56 

8.3785 

9.  i5i2 

o.7472 

—  8.3259 

+9.8274 

+9.i685 

0.5234 

9-9939 

n57 

8.o83i 

8.8457 

0.4117 

+7.  6266 

—  9.8752 

-8.7735 

0.5333 

9.9936 

n58 

8.1696 

8.9302 

0.2976 

+  7.9731 

—  9.9830 

—9-0365 

o.535 

9.9936 

1169 

8.1696 

8.9298 

0.2981 

+7-9727 

—9.9829 

—9.0362 

0.5355 

9.9936 

1160 

8.1696 

8.9177 

O.3l42 

+  7-9436 

—9.9747 

—  9.oi94 

o.5376 

9.9935 

1161 

8.3o63 

9.o63o 

o.o654 

+  8.22I7 

—0.0242 

—  9.  l52I 

o.5389 

9.9934 

1162 

8.4947 

9.232^ 

0.7947 

—  8.46oo 

+9.87oi 

+  9.2205 

o.557. 

9.9929 

n63 

8.3170 

9.0349 

o.  1268 

+8.2I83 

—  0.0190 

—  9.i757 

o.5765 

9.9922 

n64 

8.1871 

8.893o 

0.3449 

+7.9255 

-9.9544 

—  9.0242 

o.588i 

9.99i8 

u65 

8.1610 

8.85i2 

o.5593 

—7.74i5 

—8.2480 

+  8.88l8 

o.5935 

9.99i5 

1166 

8.1714 

8.8632 

0.5709 

—7.8205 

+8.6857 

+8.9485 

o.6oi5 

9.9912 

1167 

8.i595 

8.85o6 

0.5590 

—  7.7488 

—  8.2878 

+8.8893 

0.6023 

9.9912 

1168 

8.3621 

9.0326 

o.  i3oi 

+8.2632 

—  o.oi76 

—  9.22O9 

0.6220 

9.9903 

1169 

8.1798 

8.8484 

o.4o32 

+7-76l9 

—9.8890 

—  8.9o37 

0.6238 

9.9902 

1170 

8.1496 

8.8i5i 

0.4741 

+6.9946 

—  9.7ooi 

—  8.i696 

0.6269 

9.9901 

1171 

8.1794 

8.8447 

o.5539 

-7.7396 

—8.6929 

+8.8849 

0.62-70 

9.9901 

1  172 

8.1498 

8.8i5i 

o.474i 

+6.9947 

—  9.7001 

—  8.169^ 

0.62-70 

9.9901 

n73 

8.44i2 

9.1047 

9.9436 

+8.3752 

—  0.0260 

—  9.2606 

0.6288 

9.9900 

n74 

8.2489 

8.9098 

+o.32i5 

+8.0254 

—  9.9694 

—9.io55 

o.63i2 

9.9899 

1176 

8.  7121 

9.37io 

—9.1635 

+8.6947 

—  0.0248 

—  9.3i35 

o.633i 

9.9898 

1176 

8.3070 

8.9544 

+0.2606 

+8.i468 

—9.9948 

-9.1817 

o.644i 

9.9893 

1177 

8.7634 

9.4072 

—0.2747 

+8.7489 

—  0.0226 

—  9.33o7 

o.6475 

9-989i 

1178 

8.1974 

8.8271 

+0.4354 

+7.6o67 

—9.8256 

—  8.768o 

0.6610 

9.9884 

1179 

8.2475 

8.8750 

0.5826 

—7-9477 

+9.o5i2 

+9.0609 

o.663o 

9.9883 

1180 

8.2607 

8.8860 

0.3507 

+  7.9908 

—9.9488 

—  9.0930 

o.665i 

9.988i 

1181 

8  .  4649 

9.0828 

o.  0091 

+8.39i5 

—0.0226 

—9.2965 

0.6-721 

9-9877 

1182 

8.2379 

8.8437 

+0.5555 

—7.8107 

-8.5977 

+8.954i 

0.6835 

9.987i 

n83 

8.6987 

9.3002 

—9.8553 

+8.6746 

—  0.0246 

—  9.36i3 

o.6876 

9.9868 

n84 

8.3089 

8.9099 

+o.6o83 

—  8,0911 

+9.384o 

+9.1680 

0.6880 

9.9868 

n85 

8.2730 

8.8637 

o.5747 

—7-9429 

+8.84i4 

+9.o654 

o.6977 

9.9862 

1186 

8.3422 

8.93o3 

0.6217 

—  8.i567 

+9.4812 

+  9.2124 

0.7001 

9.986o 

1187 

8.2263 

8.8n4 

o.5o33 

—7.1755 

—  9.5392 

+8.3498 

O.7O29 

9.9868 

1188 

8.2387 

8.8220 

o.44o4 

+7.6114 

—  9.8i29 

-8.775i 

o.7o46 

9-9857 

1189 

8.3359 

8.9i35 

0.6112 

—8.1266 

+  9.4o74 

+9.1984 

o.7ioo 

9.9863 

1190 

8.3479 

8.9221 

0.6168 

—  8.i52i 

+9.4489 

+9.2i5i 

o.7i3i 

9.9861 

1191 

8.2734 

8.8390 

o.553o 

—7.8328 

—8.7372 

+8.9783 

O.72II 

9.9845 

1192 

8.2723 

8.8109 

o.5i26 

—7.4267 

—  9.4664 

+8.5973 

o.746i 

9.9826 

n93 

8.324o 

8.85oo 

o.566i 

—7.9581 

+  8.3222 

+9-°897 

o.7577 

9.98i6 

1194 

8.4027 

8.9i3o 

0.3097 

+8.2006 

—  9.9721 

—9.26-78 

O.7722 

9.9802 

n95 

8.5543 

9.o633 

o.o543 

+8.47^9 

—  o.oi55 

—9.392-7 

o.7733 

9.9801 

1  196 

8.32oo 

8.8289 

o.546o 

-7.8374 

—8.9796 

+8.9886 

o.7734 

9.9801 

1197 

8.3309 

8.8337 

0.4112 

+7.8878 

-9.8743 

—9.o337 

o.779o 

9-9796 

1198 

8.3i53 

8.8n4 

0.4496 

+  7.6o85 

-9.7873 

—  8.776i 

o.785o 

9-979° 

1199 

8.4ii? 

8.9053 

o.3i82 

+8.1998 

—9.96-71 

-9.273i 

o.7873 

9-9787 

1200 

+8.5786 

—  9.0718 

+  O.O32I 

+8.5o44 

—  o.oi54 

—9.4u3 

—  o.7878 

—  9-9787 

448 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.,     Annual 
Jan.  1,  1850.       Variation. 

A.      771.            *. 

s. 

0              '                  // 

" 

I2OI 

6608 

Sagittarii,          ft1 

9* 

19  ii  5o.53 

+4.328 

1  34  44    8.0 

~  5-99 

I2O2 

6610 

Sagittarii,          /32 

4 

12  21.92 

4.33i 

i35     4  36.4 

6.o3 

I2O3 

6612 

57  Draconis,           d 

3 

12  3o.45* 

o.o4i 

22  36     8.3* 

6.3i 

I2O4 

6619 

44  Sagittarii,          pl 

5 

12  58.i8* 

3.489 

108     7  28.7* 

6.34 

1205 

6622 

Sagittarii,           a 

4 

i3  29.33 

4.179 

i3o  53  34.4 

6.06 

I2O6 

6623 

i  Cygni,                K 

4 

i3  37.94* 

1.389 

36  54  23.  9* 

6.42 

I2O7 

6644 

3  1  Aquilae,               b 

5 

17  49.06* 

2.865 

78  22  i8.3 

7.37 

1208 

6646 

3o  Aquilae,               d 

H 

17  56.  o5* 

3.027 

87  10  48.6* 

6.80 

1209 

6649 

Telescopii,        p 

4 

18  23.67 

+4.901 

i45  24  42.2 

6.53 

1210 

665o 

60  Draconis,           r 

41 

18  24.87 

—  i  .096 

16  55  30.9 

6.81 

I2II 

6662 

58  Draconis,           ir 

4 

19  53.25* 

+0.332 

24  34  26.3* 

6.87 

1212 

6674 

6  Vulpeculae,         a 

4 

22    27.88 

2.495 

65  38     6.3* 

6.97 

I2l3 

6690 

6  Cygni,                (3 

3 

24    40.37* 

2.420 

62    21        7.2* 

7.26 

1214 

6697 

10  Cygni,                i1 

5 

25  55.  37* 

i.5i6 

38  35   i7.5* 

7-47 

I2l5 

6701 

38  Aquilae,              p 

41 

26  45.67* 

2.934 

82  56     8.4* 

7.32 

1216 

6703 

37  Aquilae,              k 

5 

26  5i.35 

3.3i3 

100  52  59.9 

7.4i 

1217 

6706 

52   Sagittarii,          A2 

41 

27  34.39* 

3.663 

Il5     12     33.7* 

7-48 

1218 

67i3 

39  Aquilae,               K 

4 

28  49.24 

3.233 

97  21   23.  o 

7.60 

1219 

67i5 

4  1  Aquilae,               i 

5 

28  57.70 

3.  in 

91   36  52.7 

7.60 

I22O 

6729 

44  Aquilae,                a 

5 

3i  47-36* 

2.965 

84  56  24.7* 

7.83 

1221 

6734 

1  3  Cygni,                6 

4 

32  25.  18* 

+  i  .611 

4o     7  27.6* 

8.08 

1222 

6735 

6  1   Draconis,            <r 

5 

32  38.53* 

—  o.  107 

20  35   38.i* 

6.06 

1223 

6739 

5  Sagittse,              a 

4 

33  23.  61 

+2.685 

72   19  36.6 

7.98 

1224 

6740 

12  Cygni,                <j> 

4 

33  27.16 

2.368 

60    II     22.3 

8.01 

1225 

6742 

55  Sagittarii,          e2 

5 

33  56.io* 

3.44o 

106  28   i3.8* 

8.o3 

1226 

6744 

6  Sagittae,             /? 

5 

34  18.82 

2.697 

72    52       1.2 

8.o4 

1227 

6748 

Cygni, 

5 

35  i8.i5 

1.348 

35  22  34.5 

8.10 

1228 

6771 

1  5  Cygni, 

5 

38  52.09 

2.166 

53     o  16.7 

8.47 

1229    6772 

5o  Aquilae,               y 

3 

39     7.67* 

2.857 

79  44  54-7* 

8.42 

1230 

6779 

1  8  Cygni,                6 

31 

4o  17.23* 

1.876 

45  i3  58.7* 

8.5o 

I23l 

6780 

Cygni* 

5 

4o  17.99        i.iSfi 

32    20    22.  P 

8.5o 

1232 

6783 

7  Sagittae,              6 

4 

4o  42.16  ,     2.684 

71  49  54-7 

8.58 

1233 

6784 

17  Cygni,                x 

5 

4o  44.09  .!     2.275 

56  37     4.1 

8.02 

1234 

6794 

8  Sagittaj,              f 

5           42  19.25 

2.668 

71    i3  48.o 

8.73 

1235 

6801 

Pavonis,             K 

4 

43    8.80 

7-094 

i63   17  48.4 

8.64 

1236 

6802 

53  Aquilae,               a 

il 

43  27.78* 

2.925 

81   3!   26.3* 

9-i3 

I23? 

6811 

55  Aquilae,               77 

4          44  49.86 

3.o6o 

89  22  3i.6 

8.82 

1238 

6812 

Sagittarii,           i 

41        44  54.98 

4.179 

i32    i5  26.6 

8.82 

1239 

6817 

Cygni,                      |  5 

45  28.47* 

2.o58 

49  46  45.o* 

8.91 

I24o 

6825 

59  Aquilae,               f 

5 

46  58.  67* 

2.912 

81    55  23.2* 

8.97 

1241 

6827 

1  3  Vulpeculae, 

5 

47     5.23 

2.552 

66   18  28.5 

9.11 

1242 

6832 

59  Sagittarii,          b 

5 

47  44.09* 

3.696 

117  33  44-7* 

9.09 

1243 

6833 

Co  Aquilae,               ft 

31 

47  56.66* 

2.952 

83  57  5i.5* 

8.56 

1244 

6849 

22  Cygni, 

5 

5o  30.19 

2.i44 

5i   54  32.9 

9.3o 

1245 

685i 

21   Cygni,                7j 

5 

5o  4i.io 

2.253 

55   18  47.0 

9.27 

1246 

6857 

Cygni, 

5 

52        1.19 

2.081 

5o     i   58.3 

9.42 

1247 

6858 

12  Sagittse,              y 

41 

52       5.22 

2.669 

70  54  4i  .0 

9.5o 

1248 

6866 

1  4  Vulpeculae, 

5 

52  44.48 

2.576 

67  18   16.4 

9-45 

1249 

6870 

62  Sagittarii,          c 

41 

53  25.57* 

3.7o5 

118     7  19.1* 

9.  58 

I25o 

6873 

Pavonis,             6 

4 

19  53  56.99 

+5.969 

i56  33  18.1 

-  8.5o 

TABLE    XXX. 


449 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b                c 

d 

a' 

*'                 c' 

d' 

1201 

+8.46i5 

-8.95o7 

+0.6366 

—8.3090 

+9.6678 

+9.3365 

—  o.79i3 

-9.9783 

1202 

8.467i 

8.953o 

o.638o 

—  8.3172 

+9.5643 

+9.3422 

0.7943 

9.9780 

1203 

8.7321 

9.2171 

8.2672 

+8.6974 

—0.0188 

-9.4682 

o.7952 

9-9779 

1204 

8.34i6 

8.8236 

0.5424 

—7.8345 

—9.0682 

+8.9886 

o.7978 

9.9776 

I2O5 

8.444o 

8.9227 

0.6201 

—8.2600 

+9.4672 

+9.3i46 

0.8006 

9.9773 

1206 

8.5448 

9.0226 

o.i4o4 

+8.4477 

—  o.oo83 

—  9.4o23 

0.8016 

9-9772 

1207 

8.3555 

8.8074 

o.4489 

+7.6599 

—9.7890 

—  8.8269 

0.8248 

9-9745 

1208 

8.  3476 

8.7988 

0.4784 

+7.o395 

—9.6811 

—  8.2161 

0.8264 

9.9744 

1209 

8.5954 

9.0439 

+0.6900 

—  8.6110 

+9.  7236 

+9.44n 

0.8278 

9-974i 

1210 

8.8855 

9.3339 

—  0.0287 

+8.8663 

—  0.0118 

—  9.5o65 

o.8279 

9.9741 

I2II 

8.7384 

9.  1780 

+9.6088 

+8.6972 

—  o.oi5o 

—  9.4922 

o.8357 

9.9731 

1212 

8.4m 

8.8357 

o.3986 

+8.0266 

—9.8929 

—  9.  1621 

o.8489 

9-97i3 

1213 

8.4342 

8.8462 

0.3834 

+8.1008 

—  9.9123 

—  9.2242 

o.8599 

9.9697 

1214 

8.5926 

8.9977 

0.1794 

+8.4856 

—9-9992 

—9.4667 

o.8659 

9.9687 

I2l5 

8.3949 

8.7953 

0.4649 

+7-4849 

-9.7364 

—8.6676 

o.8699 

9.9681 

1216 

8  .4ooo 

8.7998 

0.6198 

—7.6760 

-9.3993 

+8.8442 

0.8704 

9.9680 

1217 

8.4390 

8.8349 

0.5628 

—8.0683 

+  7.4160 

+  9.2OO9 

0.8738 

9.9676 

1218 

8.4049 

8.7940 

0.6093 

—7.5i23 

—  9.494o 

+8.6848 

o.8796 

9.9666 

1219 

8.4021 

8.7905 

0.4921 

—6.8621 

—  9.6io3 

+8.0280 

o.88o3 

9.9664 

1220 

8.4166 

8.7898 

0.4716 

+  7.3620 

-9.7io5 

—8.5364 

o.8932 

9.9642 

1221 

8.6o85 

8.9784 

+0.2072 

+8.49i9 

—  9.9926 

—9.4772 

o.896o 

9.9637 

1222 

8.8724 

9.2412 

—9.3023 

+8.8437 

—  0.0068 

—  9.566o 

o.8969 

9.9635 

1223 

8.443o 

8.8078 

+0.4281 

+  7.9262 

—  9.84o2 

—  9.o8o3 

o.9oo3 

9.9629 

1224 

8.4839 

8.8484 

0.3743 

+8.i8o3 

—  9.92o6 

—  9.2948 

o.9oo5 

9.9629 

1225 

8.4425 

8.8o46 

o.5357 

-7.8961 

—  9.i937 

+9.o53o 

0.9027 

9.9626 

1226 

8.4457 

8.8o58 

o.43o2 

+  7.9149 

—9.8355 

—  9.0713 

o.9o43 

9.9621 

1227 

8.6676 

9.0226 

0.1297 

+8.6790 

—9.9987 

—9.6177 

0.9086 

9.9613 

1228 

8.5429 

8.8798 

0.3336 

+8.3223 

—  9.9610 

—  9.4007 

0.9235 

9.9683 

I229 

8.4533 

8.7890 

o.455o 

+7.7o36 

—9.7698 

—8.8727 

0.9246 

9.9680 

1230 

8.5998 

8.9297 

0.2717 

+8.4475 

-9.9769 

—9-4748 

0.9293 

9.96-70 

I23l 

8.7228 

9.0526 

o.o637 

+8.6496 

—9.9989 

—  9.554o 

0.9294 

9.957o 

1232 

8.4749 

8.8028 

O.427I 

+7.9688 

—9.8417 

—  9.  1227 

0.9310 

9.9667 

1233 

8.53u 

8.  .8588 

o.3567 

+8.2717 

—9.9347 

—  9.3695 

0.9311 

9.9666 

1234 

8.4829 

8.  .8028 

0.4260 

+7.9906 

—9.8468 

—  9.i428 

o.9375 

9.9552 

1235 

9.0039 

9..3i98 

0.8602 

—8.9862 

+9.8679 

+9.6198 

0.9407 

9.9646 

1236 

8.4684 

8.7828 

o.46i  i 

+  7.6369 

—9.7496 

—8.8082 

0.9420 

9.9642 

I237 

8.4690 

8.7768 

o.4854 

+6.5o64 

—9.6471 

—7.6824 

o.9473 

9.9629 

1238 

8.6000 

8.9074 

0.6191 

-8.4277 

+9.45o3 

+9-473i 

o.9476 

9.9628 

I239 

8.5886 

8.8934 

o.3i34 

+8.3986 

—9.9690 

—9.4676 

o.9497 

9.9623 

I24o 

8.48i5 

8.7791 

0.4626 

+  7.6292 

—9.7446 

—8.8009 

o.9555 

9.9609 

I24l 

8.5i58 

8.8129 

o.4o6o 

+8.1198 

—9.8786 

-9.2677 

o.9559 

9.9608 

1242 

8.5323 

8.8264 

o.5673 

—8,1976 

+8.  ,444o 

+9-32I4 

o.9583 

9.9601 

1243 

8.4832 

8.7763 

0.4690 

+7.  ,6060 

—9.7204 

-8.6787 

0.9691 

9.9499 

1244 

8.5942 

8.8754 

o.33o8- 

+8.3844 

—9.9486 

—9.4666 

0.9686 

9-9474 

1245 

8.5758 

8.856i 

0.3524 

+8.33io 

—  9.9349 

—9.4221 

0.9691 

9.9473 

1246 

8.6m 

8.8854 

o.3i83 

+  8.4189 

—  9.954i 

—9.4795 

o.9739 

9.9459 

124? 

8.52o4 

8.7943 

0.4262 

+8.o35o 

-9-8444 

—9.1866 

o.974i 

9.9469 

1248 

8.5332 

8.8o4i 

O.4lI2 

+8.1196 

—9.8696 

—  9.2606 

o.9765 

9.9462 

1249 

8.555: 

8.823o 

0.6681 

—8.2286 

+8.6066 

+9.35oo 

o.9789 

9.9446 

1260 

+8.9027 

—9.1681 

+0.7620 

—8.8662 

+9.8019 

+9.64n 

—  o.98o7 

—  9.9440 

FF 


450 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation.             j  Mag. 

l 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

A.     m.         s. 

s. 

0            /                // 

1261 

6877 

Sagittarii, 

5 

19  54  48.47* 

-f  3.84o 

122    28    22.1* 

—    9-54 

1262 

6879 

1  5  Vulpeculae, 

5 

54  55.  4o* 

2.469 

62    39    28.0* 

9.72 

i253 

6882 

Vulpeculae, 

5 

55  23.47 

2.543 

65  36  44.2 

9.72 

1254 

GgoS 

64  Draconis,          e 

5 

59  52.  4i 

o.65i 

25  35  54.4 

10.  OO 

1255 

6926 

67  Draconis,          p 

4 

2O       2       7.69* 

o.3oo 

22  33  14.1* 

10.23 

1256 

6932 

66  Draconis, 

5 

3     9.17* 

0.964 

28  26  19.  7* 

10.32 

i257 

6934 

65  Aquilae,              0 

at 

3  33.76* 

3.io3 

91   16  44.8* 

IO.32 

1208 

6937 

28  Cygni,               £2 

5 

3  5i.43* 

2.228 

53  35  56.  7* 

10.43 

I259 

6952 

67  Aquilae,              p 

5 

7  20.  25 

2.779 

75  i5  21.9 

10.66 

1260 

6959 

Cygni, 

5 

8  21.19 

1.671 

38  59  10.  o 

io.65 

1261 
1262 

6965 
6966 

3  1   Cygni,               o2 
Vulpeculae, 

4 
5 

8  54.58* 
8  54.68 

i  .890 
2  .  54o 

43  42  4o.6* 
64  5i  43.2 

10.76 
10.70 

1263 

6972 

5  Capricorni,        a1 

4 

9  19.76* 

3.334 

102  58     4.0* 

io.75 

1264 

6973 

23  Vulpeculae, 

4t 

9  33.36 

2.485 

62  38  32.4 

10.  80 

1266 

6974 

6  Capricorni,       a2 

3 

9  43.67* 

3.339 

io3     o  20.  7* 

10.  79 

1266 

6976 

33  Cygni, 

4* 

9  54.48* 

i  .4oi 

33  53  23.o* 

10.  81 

1267 

6979 

24  Yulpeculae, 

5 

IO    22.  II 

2.569 

65  47  i4-o 

10.82 

1268 

6983 

32  Cygni, 

4* 

10  49.97* 

1.854 

42  44  39.2* 

10.87 

1269 

6991 

8  Capricorni,        v 

5 

12    2O.  3o* 

3.337 

io3  i3  36.  7* 

10.9^ 

1270 

6995 

9  Capricorni,       j3 

3* 

12    34.69* 

+  3.38o 

io5  i5     3.4* 

II  .01 

1271 

6999 

Ursae  Minoris,  A 

5 

i3     1.76* 

—53.184 

I        8    22.0* 

II  .02 

1272 

7004 

Pavonis,            a 

2 

i3  45.  o3 

+  4.807 

i47  12  35.3 

II  .02 

I273 

7oo5 

i  Cephei,             «: 

4* 

i3  5o.39* 

—   1.862 

12  44  34.  7* 

II  .07 

1274 

7022 

37  Cygni,               y 

3 

16  5o.8i* 

+  2.i53 

5o  i3   i5.4* 

ii.3o 

1275 

7027 

Cygni, 

5 

17  25.78 

2.  126 

49    27       4.6 

11.32 

1276 

7029 

39  Cygni, 

5 

17    52.25 

2.395 

58  17  26.5 

ii  .40 

1277 

7o3i 

10  Capricorni,        TT 

5 

18  43.79* 

3.446 

i  08  4i  57.1* 

n.46 

1278 

7042 

ii  Capricorni,       p 

5 

20    17.88* 

3.433 

108   18  19.4* 

H.55 

1279 

7o58 

69  Aquilae, 

5 

21    48.62 

3.!45 

93    22    49.  I 

n.65 

1280 

7067 

4  1  Cygni, 

4i 

23   15.99 

2.45o 

60     7  46.2 

ii.  75 

1281 

7o85 

45  Cygni,               tf 

5 

25  24.79 

1.858 

4i  33     2.5 

ii  .91 

1282 

7088 

2  Delphini,           * 

4 

26     2.66* 

2.868 

79  12  ii.  2* 

11.98 

1283 

7091 

46  Cygni,               w3 

5 

26  40.95 

i.85i 

4i   17     i.5 

ii  .95 

1284 

7096 

T         1* 

Indi,                   a 

3 

26  59.94 

4.264 

i37  48  34.o 

12  .06 

1285 

7098 

2  Cephei,             0 

5 

27     3.34* 

i  .020 

27  3o  32.i* 

11.99 

1286 

7106 

Pavonis,            v 

5 

28     7.79 

5.642 

i57  17     o.3 

II  .96 

1287 

7107 

4  Delphini,           C 

5 

28  17.73* 

2.807 

75  5o  23.9* 

12.  l3 

1288 

7121 

6  Delphini,           J3 

4 

3o  3o.86* 

2.8i3 

75  55  25.o* 

12.23 

1289 

7122 

71   Aquilae, 

5 

3o  35.24* 

3.io4 

91   37  3i.i 

12  .27 

1290 

7129 

Pavonis,            /3 

3 

3i   22.68 

5.524 

i56  44     9.6 

12.29 

1291 

7i34 

1  5  Capricorni,        v 

5 

3i   3o.26* 

3.426 

108  39  45.9* 

12.36 

1292 

7i37 

8  Delphini,           6 

41 

3i   39.29 

2.833 

77     12    26.7 

12.36 

1293 

7149 

9  Delphini,           a 

H 

32  40.26* 

2.791 

74  36  49.4* 

12.  4l 

129^ 

7i65 

Pavonis,            a 

41 

35     o.5i 

5.823 

169   19     i.5 

12.63 

1295 

7171 

5o  Cygni,                a 

i 

36  19.12* 

2  .  o4^ 

45  i5  12.4* 

12.64 

1296 

7i73 

ii  Delphini,           6 

4 

36  27.50* 

2.804 

75  27  36.8 

12.63 

1297 

7177 

1  6  Capricorni,       V 

4* 

37   12.  3i* 

+  3.57o 

n5  48   21.7* 

12.55 

1298 

7178 

75  Draconis, 

si 

37    25.20* 

—  3.382 

9     5  4i.9* 

12.  72 

1299 

7184 

Ursae  Minoris, 

5 

3?  47-97 

—41.226 

i   20     4-6 

12.  75 

i3oo 

7196 

2  Aquarii,              e 

41 

20  39  33.  i3* 

+  3.259 

IOO       2    28.6* 

—12.84 

TABLE    XXX. 


451 


CATALOGUE   OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

Y-  g    i  -:?.-( 

b 

c 

d 

a' 

W 

c' 

d' 

I25l 

-f-8.5792 

—  8.84o8 

+o.58i7 

—  8.3091 

+  9.O2o4 

+9.4n4 

—  o.9837 

—  9.943i 

1262 

8.5572 

8.8i83 

o.39i7 

48.2i93 

—  9.8967 

—  9.344o 

0.9841 

9.9429 

1253 

8.548o 

8.8o7o 

o.4o48 

+8.I638 

-9.879o 

—  9.2993 

o.9857 

9.9425 

1254 

8.8869 

9.  1260 

9.8i49 

+8.8421 

—  9  .9862 

—  9.6537 

i  .0008 

9-9377 

1255 

8.9460 

9.i753 

9.4738 

+8.9114 

—  9.983i 

—  9.67i3 

i  .0081 

9.9352 

1256 

8.8553 

9.0801 

9-9776 

+8.7994 

—  9.9832 

—9.  6533 

i  .on4 

9.934o 

I257 

8.5345 

8.7576 

0.4908 

—  6.8775 

—  9.6i8i 

+8.o535 

1.0127 

9.9336 

1258 

8.6296 

8.85i4 

o.3474 

+8.4o3o 

-9.9336 

—  9.4848 

i.oi36 

9.9332 

1259 

8.5607 

8.7677 

0.4428 

+7.9663 

—  9.8o34 

—  9.12-79 

1.0244 

9.9292 

1260 

8.75o5 

8.9532 

0.2230 

+8.64io 

—  9.9692 

—  9.6i59 

I  .O275 

9.9280 

1261 

8.7u4 

8.9118 

0.2759 

+8.57o4 

—  9.9597 

—  9.586o 

1.0292 

9.9273 

1262 

8.5941 

8.7945 

o.4o48 

+  8.2223 

-9.8764 

—  9.3552 

I  .O292 

9.9273 

1263 

8.5634 

8.762o 

O.5225 

—  7.9i44 

-9.3679 

+9-°792 

i.o3o4 

9.9268 

1264 

8.6o43 

8.8020 

o.3956 

+8.2666 

—  9.8885 

—  9.  39i2 

i.oSn 

9.9266 

1265 

8.5646 

8.76i6 

0.5226 

—  7.9i69 

—  9.3672 

+9.o8i7 

i.o3i6 

9.9264 

1266 

8.8o75 

9.oo37 

o.i435 

+8.7267 

-9.9742 

—  9.649i 

I  .O322 

9.9261 

1267 

8.5952 

8.7895 

o.4o89 

+8.2081 

—  9.87oo 

—9.3442 

i.o335 

9.9256 

1268 

8.7249 

8.9i72 

0.2680 

+8.59o8 

—  9.9602 

—  9.5986 

I.o349 

9.9250 

1269 

8.  5727 

8.7588 

o.523o 

—  7.9322 

—  9.363i 

+  9.o966 

i.o393 

9.9232 

1270 

8.5773 

8.7624 

+0.5284 

—7-9973 

-9.2978 

+9.i578 

i  .o4oo 

9.9229 

1271 

o  .  2644 

o.4477 

—  i  .7254 

+0.2643 

—  9.9267 

—  9.739o 

I.o4i3 

9.9223 

1272 

8.'83i4 

9.0117 

+o.68i5 

—  8.756i 

+9.6689 

+9.6658 

i.o434 

9.9214 

1273 

9.2218 

9.4017 

—  o.2699 

+  9.2IIO 

-9.9588 

—  9.73o6 

1.0437 

9.9213 

1274 

8.6882 

8.8558 

4-0.3325 

+  8.4943 

—  9.9358 

—  9.556o 

1.0622 

9.9I75 

1275 

8.6947 

8.8599 

o.3276 

+8.5o77 

—  9-9377 

—  9.5645 

i.o538 

9-9l67 

1276 

8.6469 

8.8io3 

0.3783 

+8.3676 

—  9.9o45 

—  9.4734 

i.o55o 

9.9162 

1277 

8.6026 

8.7625 

o.5369 

—8.1086 

—  9.i688 

+  9.2611 

i  .0574 

9.9150 

1278 

8.6o59 

8.7594 

0.5356 

—  8.  I029 

—  9.i9o3 

+9.2564 

i  .0616 

9«9i3o 

1279 

8.588i 

8.7356 

O.4962 

-7.3587 

-9.5857 

+8.534o 

i  .0657 

9.9109 

1280 

8.653i 

8.7948 

0.3888 

+8.35o4 

—  9.8922 

—  9.4646 

i.o695 

9.9090 

1281 

8.775i 

8.9082 

0.2685 

+8.6492 

—  9.9496 

—  9.647o 

i  .0751 

9.9060 

1282 

8.6062 

8.7368 

o.4572 

+  7.8788 

—  9.76u 

—  9.o47i 

1.0767 

9.9051 

1283 

8.78o6 

8.9o87 

0.2670 

+8..  6565 

—  9.9489 

—  9.652o 

1.0784 

9.9042 

1284 

8.7737 

8.9006 

0.6286 

—8.6435 

+9.4804 

+9.6467 

1.0792 

9.9o38 

1285 

8.9365 

9.o63i 

0.0060 

+8.8843 

-9.9595 

—9.7250 

i.o793 

9.9o37 

1286 

9.0169 

9.i393 

0.7497 

—  8.98i9 

+9.755o 

+9-7447 

i  .0820 

9.9022 

1287 

8.6i75 

8.7392 

o,4474 

+8.0060 

-9.7896 

—  9.i687 

1.0824 

9.9019 

1288 

8.6229 

8.7359 

o.448o 

+8.0089 

-9.7878 

—  9-I7I7 

i.o879 

9.8987 

1289 

8,  6  i  oo 

8  .7227 

o.49i^ 

—  7.  062-7 

—  9.6i42 

+8.2387 

i.  0881 

9.8986 

1290 

9,.oi52 

9.  1248 

0.742^ 

—  8.9784 

+  9.7436 

+9.75io 

I  .O9OI 

9.8975 

1291 

8.6355 

8.7446 

o.5349 

—  8.i4o6 

—  9.  2OOO 

+9.2933 

i  .o9o4 

9.8973 

1292 

8.6233 

8.73i9 

0.4520 

+7.9686 

—  9.7766 

-9.i337 

i  .o9o7 

9.897o 

1293 

8.63o7 

8.7353 

0.4443 

+8.o545 

—9-7973 

—  9.2l47 

I  .O932 

9.8955 

1294 

9.0-72/1 

9.  i679 

o.7659 

—9.0435 

+9.76o7 

+9-7676 

i.o988 

9.892O 

I295 

8.772I 

8.8625 

o.  3ioo 

+8.6i97 

-9.93i8 

—  9.6472 

i  .  1018 

9.8900 

1296 

8.638o 

8.7278 

0.447^ 

+8.o378 

—  9.7888 

—  9.1997 

I  .  1022 

9.8898 

1297 

8.67i2 

8.7582 

4-0.5528 

—8.3ioo 

—8.7308 

+9.44o5 

i  .  io39 

9.8886 

1298 

9.4272 

9.5i34 

—  o.  529c 

+  9.4218 

—  9.9202 

—  9-7967 

i  .  i  o44 

9.  8883 

1299 

0.2599 

0.3443 

—  i.6i5a 

+0.2598 

—  9.893i 

—  9.8o3i 

i.io55 

9.88-76 

i3oo 

+8.6377 

—  8.7i56 

4o.5l22 

—  7.8791 

-9.4676 

+9.o485 

—  i  .io93 

—  9.885o 

452 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.  . 
Jan.  1,  1850. 

Annual 
Variation. 

h.     m.        s. 

s. 

0              /                // 

l3oi 

7200 

12  Delphini,            y 

4 

20  39  42.  o5 

+2.  785 

74  24  44.9 

—  12.72 

1302 

7201 

3  Aquarii, 

4 

39  49-25* 

3.i73 

95  34  24.3* 

12.85 

i3o3 

7204 

53  Cygni,                e 

3 

4o     8.55 

2.426 

56  35  20.4 

i3.23 

i3o4 

7207 

Microscopii,       a 

4* 

4o  35.  59 

3.79.0 

124   19  5i.i* 

12.78 

i3o5 

7213 

54  Cygni,                2, 

5 

4i  34.o6 

2.333 

54     3  27.9 

i3.o6 

i3o6 

7216 

Cephei, 

5 

4i  37.72* 

1.496 

32  57   24.9* 

12.81 

iSoy 

7220 

3  Cephei,              ij 

3* 

4a  i3.97* 

1.232 

28  44  33.6* 

i3.86 

:3o8 

7228 

Indi,                   p 

4 

43     2.81 

4.757 

i49     o   54.8 

1  3.  09 

1309 

?239 

6  Aquarii,              p. 

4* 

44  33.56* 

3.246 

99   32   33.3* 

i3.i7 

i3io 

7260 

Octantis,             a 

4* 

46  23.  o7 

7-6i4 

167   35    17.5 

12.73 

i3u 

7253 

57  Cygni, 

5 

47  56.44* 

2.I2O 

46   10  43.i* 

13.43 

I3l2 

7256 

32  Vulpeculae, 

41 

48  10.  o5* 

2.557 

62   3o  36.5* 

13.47 

i3i3 

7277 

58  Cygni,                v 

4 

5i  35.  o4* 

2.234 

49  24  28.8* 

i3.68 

i3i4 

7281 

Cephei, 

5 

52  16.14* 

4-i.  6o5 

33  4i    i8.5* 

13.67 

i3i5 

7291 

76  Draconis, 

5 

53     7.6i* 

—3.832 

8     i   43.  9* 

13.78 

i3i6 

7299 

Cephei, 

5 

54  i3.i7* 

—  2.422 

10     o  47.2* 

i3.82 

i3i7 

73o5 

22  Capricorni,         rj 

5 

55  5i.59* 

+  3.43o 

110  26  4o.9* 

i3.9o 

i3i8 

7333 

62  Cygni,                £ 

4 

59  28.68* 

2.180 

46  4o    5.7* 

14.17 

iSig 

7336 

6  1   Cygni, 

51 

21        O    IO.7I 

2.69I 

5i    59     8.0 

17.49 

1320 

7344 

1  3  Aquarii,              v 

5 

i  25.  o3* 

3.276 

101   58   32.7* 

14.28 

1321 

7345 

63  Cygni,               /» 

5 

I  26.08* 

2.o63 

42  57     8.7* 

14.28 

1322 

735o 

5  Equulei,             y 

5 

3     2.89 

2.923 

80  28  i  i.  i 

l4-20 

i323 

7368 

64  Cygni,  ^              $ 

3 

6  33.22* 

2.55o 

60  23     9.4* 

14.52 

i324 

7372 

7  Equulei,              d 

4* 

7  10.  57 

2.927 

80  35  52.5 

14.34 

i325 

7374 

29  Capricorni, 

5 

7  26.34* 

3.334 

io5  47  29.4* 

14.66 

i326 

?3?7 

Cephei, 

5 

7  59.o5* 

i.54a 

3o  37  44.3* 

i4.63 

I327 

738o 

8   Equulei,              a 

4 

8  i9.45*+3.oo5 

85    22     IO.I* 

i4.6i 

i328 

738i 

77  Draconis, 

5$ 

8  22.65    —  i  .o46 

12  28  59.6 

14.72 

1329 

7385 

65  Cygni,                r        5 

8  48.33* 

+  2.39I 

52   35   33.o* 

i5.23 

i33o 

7386 

4  Piscis  Aust., 

5 

8  49.7i* 

3.665 

122  47  45.8* 

i4.66 

i33i 

7398 

67  Cygni,                 a 

41 

ii   3i.69 

2.353 

5i    i3   54.3 

i4.87 

i332 

7399 

66  Cygni,                 v 

5 

ii  45.12 

2.462 

55  43  49.3 

14.91 

i333 

7407 

32  Capricorni,        b 

5 

i3  53.20* 

3.357 

107    28     12.  I* 

16.07 

1  334 

74o9 

Pavonis,              y 

3 

i3  58.  74 

5.o8i 

i56     2  27.5 

i5.73 

i335 

7416 

5  Cephei,             a 

3 

i4  59.74* 

1.439 

28     2  55.5* 

15.09 

i336 

74i8 

i  Pegasi, 

4 

i5     9.o8 

2.776 

70  5o     4-3 

16.18 

i337 

74a3 

Indi,                   y 

5 

i5  3i.42 

4.337 

i45   18   18.9 

14.96 

i338 

7428 

6  Cephei, 

5 

16  i5.o7 

I  .259 

25  45  47-3 

i5.i5 

i339 

7445 

34  Capricorni,         £ 

4 

18     5.65* 

3.443 

u3     3  27.5* 

i5.3i 

i34o 

7478 

22  Aquarii,             p 

3 

23  39.5i* 

3.i68 

96  i3  41.7* 

16.60 

i34i 

7480 

71   Cygni,                g 

5 

23  54.  95* 

2.206 

44     7     8.0* 

16.69 

i342 

7493 

8  Cephei,              p 

3 

26  4a.32* 

0.807 

20     5  49.7* 

16.69 

1  343 

7495 

Cephei, 

5 

26  52.  o3 

1.647 

3o  12     3.9 

i5.74 

1  344 

75o3 

73  Cygni,                p 

41 

28  20.60* 

2.25o 

45     4  10.  o* 

16.77 

1  345 

75o6 

39  Capricorni,        e 

5 

28  4o.46* 

+  3.375 

no     8     6.5* 

16.86 

1  346 

75io 

Cephei, 

5* 

29     6.18 

—  1.389 

10     7  48.8 

i5.96 

1  34y 

75i4 

23  Aquarii,              £ 

5 

29  45.  77* 

-f-3.202 

98  3i    26.1* 

i5.89 

1  348 

7522 

4  Pegasi, 

5 

3  i      i.o9 

3.oo4 

84  54  10.9 

i5.99 

i349 

7525 

4o  Capricorni,        y 

4 

3r  46.  4o* 

3.34i 

107  20   i3.2* 

16.02 

i35o 

7539 

4  1  Capricorni, 

5 

21   33  2-7.80 

+  3.435 

n3  56   i6.3 

—  i6.o4 

TABLE    XXX. 


453 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b            I             C 

d 

a! 

V 

c'        \         d' 

1801 

+8.6476 

-8.7249 

+0.4448 

+  8.0769 

—9.7963 

—  9.2367 

—    .1096 

—9.8847 

1302 

8.6336 

8.7105 

o.5oi  i 

—  7.62IO 

—9.5532 

+8.795o 

.1099 

9.8845 

i3o3 

8.7108 

8.  7864 

0.3794 

+8.45i6 

—9.8942 

—  9.5493 

.1106 

9.8840 

i3o4 

8.7164 

8.7903 

0.6762 

-8.4677 

+8.86i5 

+9.660-7 

.1116 

9.  8833 

i3o5 

8.7272 

8.7974 

o.3678 

+8.4968 

—  9.9022 

—9.6802 

.n38 

9.8817 

i3o6 

8.9000 

8.9700 

o.  1762 

+8.8238 

—9.9420 

-9.7355 

.n39 

9.8817 

i3o7 

8.9549 

9.0226 

o.o858 

+8.8978 

—9.9420 

—  9.7660 

.n53 

9.8807 

r3o8 

8.9271 

8.9916 

0.6773 

—  8.86o3 

+9.6294 

+9.7480 

.1171 

9.8793 

1  3  09- 

8.648i 

8.7068 

o.5io5 

-7.8677 

—9.4816 

+9.o377 

.  I2O4 

9.8769 

i3io 

9.3i37 

9-3654 

0.8816 

—  9.3o34 

+9.8042 

+9.8118 

.1243 

9.8738 

i3n 

8.7911 

8.8368 

0.3257 

+8.63i4 

—9.9179 

—9.6668 

.1276 

9.8712 

l3l2 

8.7018 

8.7467 

0.4072 

+8.366i 

—  9.8620 

—  9.4901 

.1281 

9.8708 

i3i3 

8.7764 

8.8o83 

0.3486 

+8.6898 

—  9.9064 

—9.6463 

.i35s 

9.8649 

i3i4 

8.9142 

8.9435 

+o.2o56 

+8.8344 

—9.9287 

—9.7645 

.i366 

9.8637 

i3r5 

9.6149 

9.5410 

—0.5832 

+9.6106 

-9.8929 

—  9.83i8 

.1383 

0.8621 

i3i6 

9  .  42  i  9 

9.4438 

—0.3833 

+9.4i53 

—9.8972 

—  9.83i6 

.i4o5 

9.8602 

i3i7 

8.6937 

8.7o94 

+0.5352 

—  8.2369 

—9.1909 

+9-3847 

.i437 

9.8672 

i3i8 

8  .  8  i  06 

8.8126 

0.3378 

+8.647i 

—  9.9042 

—9.8660 

.  1607 

9.8606 

i3i9 

8-7773 

8.7766 

0.3678 

+8.5668 

—  9.8904 

—9.6393 

.  1620 

9.8492 

1320 

8.6856 

8.68o3 

o.5i45 

—8.0026 

-9.445i 

+9.1692 

.1644 

9.8468 

1321 

8.8427 

8.8373 

0.3l42 

+8.7o72 

—  9.9o85 

—  9.7166 

.i544 

9.8468 

l322 

8.685i 

8.6736 

0.4645 

+  7.9041 

-9.7352 

—  9.0742 

-i574 

9.8436 

i323 

8.7463 

8.  7214 

o.4o63 

+8.44oi 

-9.  8553 

-9.5554 

.i638 

9.8367 

1324 

8.6925 

8.6652 

0.4652 

+  7.9066 

-9.7323 

—9.0768 

.1649 

9.  8355 

i325 

8.7o38 

8.6755 

0.5223 

—8.1386 

—9.  3646 

+9.2979 

.1664 

9.8349 

i326 

8.9809 

8.9506 

o.i848 

+8.9166 

—  9.  0.074 

—9.7988 

.1662 

9.8338 

I32? 

8.6900 

8.6584 

+0.4767 

+  7.6970 

—  9.688o 

—8.7717 

.1669 

9.833i 

i328 

9.3539 

9.  3222 

—  0.0173 

+9.3435 

—9.8800 

—9.  8544 

.16-70 

9.833o 

1320 

8.7895 

8.7561 

+o.3758 

+8.6730 

-9.8793 

—9.6491 

.16-78 

9.832i 

i33o 

8.7649 

8.73i4 

o.563o 

—8.4986 

+7.6062 

+9.6993 

.i678 

9.8321 

i33i 

8.8os3 

8.7585 

0.3711 

+8.6990 

-9.8798 

—  9.6670 

.I725 

9.8266 

i332 

8.7774 

8.7328 

o.  3910 

+8.528o«i 

—9.8668 

—  9.6212 

.  1729 

9.8260 

i333 

8.7i87 

8.6659 

o.525o 

—8.1961 

—9.3322 

+9.35i7 

.1766 

9.8216 

i334 

9.089-7 

9  .o366 

0.7045 

—  9.0606 

+9.6329 

+9-8353 

.1766 

9.8213 

i335 

9.0277 

8.97o7 

o.  i5n 

+8.9735 

—9.8964 

—  9.8219 

.i783 

9.8191 

i336 

8.725i 

8.6675 

o.44i6 

+8.24i3 

—9.7967 

—9.3927 

.1786 

9.8188 

i  SSy 

8.9456 

8.8866 

0.6376 

—8.8606 

+9.4679 

+9.7920 

.1792 

9.8180 

i338 

9.0640 

9.0022 

0.0988 

+9.0186 

—9.8914 

-9.  8327 

.1804 

9.8164 

i339 

8.74i3 

8.6724 

0.5366 

—8.3342 

—9.1617 

+9.474i 

.i834 

9.8123 

i34o 

8.7i65 

8.6262 

o.5ooo 

-7.75i8 

—9.6699 

+8.9263 

.  1922 

9.7997 

i34i 

8.87i6 

8.78o3 

0.3429 

+8.7276 

—9.8774 

—9-7464 

.1926 

9.7991 

1  342 

9.1824 

9.o8o3 

9.9067 

+9.1662 

—9.8642 

—9.8673 

.1968 

9.7926 

1  343 

9.0172 

8.9i44 

0.2168 

+8.9538 

-9.8773 

—  9.83i5 

.1971 

9.7921 

1  344 

8.87o9 

8.7624 

0.3524 

+8.7199 

—  9.8699 

—  9.746o 

.i993 

9.7886 

i345 

8.7488 

8.639o 

+0.6278 

—8.2867 

—  9.2929 

+9.4344 

.1998 

9.7877 

1  346 

9.4768 

9.3653 

—0.1784 

+9.4700 

—  9.83i6 

—9.8913 

.2OO4 

9.7867 

1  34? 

8.7279 

8.6i38 

+o.5o4s 

—7.8988 

—9.6299 

+9.0701 

.2013 

9.7860 

1  348 

8.7266 

8.6o76 

0.4769 

+7.6762 

—9.6866 

—8.8496 

.2032 

9.7819 

1  349 

8.7461 

8.6241 

0.6214 

—  8.2203 

—9.3-706 

+9.3762 

.2042 

9.7800 

i35o 

+8.7674 

—  8.6387 

+0.5346 

—8.3766 

—9.1889 

+9.6127 

—     .2O66 

-9.7768 

454 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.        s. 

s. 

0             /                    // 

ir 

i35i 

7542 

9  Cephei, 

5 

21   33  53.84* 

-J-I  .609 

28  35  36.i* 

—  16.14 

i35a 

?543 

43  Capricorni,         K 

5 

34  16.47* 

3.364 

io9  32  49.9* 

16.16 

i353 

7557 

9  Piscis  Aust.,      t 

4i 

36     0.02* 

3.6oo 

123    42    25.2* 

16.14 

1  354 

756o 

80  Cygni,                Tr1 

tt 

36  46.26* 

2.122 

39  29  36.  7* 

16.26 

i355 

7661 

8  Pegasi,               e 

8i 

36  49.11* 

2.961 

80  48  37.o* 

16.29 

i356 

7567 

9  Pegasi, 

4* 

37  24.  73 

2.844 

73  20     4.7 

16.36 

i357 

7568 

78  Cygni,                p.1 

5 

37  26.08* 

2.673 

61   55  58.3* 

16.08 

i358 

757i 

10  Pegasi,               K 

4 

37  61.26 

2.712 

65     2  32.6 

i6.35 

i359 

758o 

49  Capricorni,         J 

|Hr 

38  45.  3o* 

3.323 

106  48  18.6* 

16.12 

i36o 

7583 

10  Piscis  Aust.,      d 

5 

38  55.  4i 

3.547 

121   35  24.2* 

16.39 

i36i 

7588 

ii   Cephei, 

4* 

39  42.86* 

0.908 

I9    22    43.0* 

16.47 

i36a 

7595 

10  Cephei,              v 

41 

4i     7.53* 

i.73i 

29  34  12.4* 

16.46 

1  363 

7597 

78  Draconis, 

5 

4i   i3.o3* 

0.762 

18    22        1.6* 

16.47 

1  364 

7598 

8  1   Cygni,                K* 

5 

4i   i5.38* 

2.2IO 

4i   22  58.4* 

16.48 

i365 

76o7 

1  4  Pegasi, 

5 

43  12.63 

2.648 

60  3  i    i9.9 

16.58 

i366 

76io 

Cephei, 

5 

44  21.68* 

i  .096 

20     32     37.1* 

16.66 

i367 

76i3 

Gruis,                y 

3 

44  49.99 

3.672 

128     4     3.9 

i6.55 

1  368 

76i8 

5  1  Capricorni,        /* 

5 

45    6.72* 

3.285 

io4  i5   18.1* 

l6.72 

1  369 

7633 

Indi,                   6 

5 

47  4o.33 

4.129 

i45  42     8.6 

16.81 

i37o 

7634 

Indi,                   K1 

5 

4y  5i.53 

4.289 

i49  43  26.4 

16.98 

i37i 

7667 

12  Piscis  Aust.,      T? 

5 

52  12.67 

3.469 

119  10  i5.o 

I7.o5 

I372 

7672 

3  1  Aquarii,              o 

5 

55  33.  3i 

3.  no 

92  52  37.2 

17.19 

i373 

7684 

Gruis,                A 

5 

57     3.47 

3.65o 

i3o  i5  55.i 

17.02 

i374 

7686 

1  6  Cephei, 

5 

57     5.37* 

o.885 

I7    32       0.0* 

17.06 

i375 

7688 

34  Aquarii,             a 

3 

58     4.64* 

3.o85 

91     2  47.4* 

17.29 

i376 

7689 

22  Pegasi,               v 

5 

58     6.89 

3.o3i 

85  4o  22.3 

17.38 

i377 

769i 

33  Aquarii,             t 

4^ 

58  19.77* 

3.262 

io4  35  42.3* 

17.26 

i378 

7692 

Gruis,                 a 

2 

58  45.o9 

3.824 

i37  4i     3.2 

17.13 

i379 

7699 

1  8  Cephei, 

5 

59    22.  29 

1.770 

27  36  34.2 

17-34 

i38o 

77oo 

17  Cephei,             £ 

5 

59  26.04* 

1.738 

26     6     6.3* 

17.42 

r38i 

77o6 

24  Pegasi,               t 

4 

22       0       1.84 

2.788 

65  23     7.4 

17.42 

r382 

772I 

27  Pegasi,               irl 

5 

,       a  35.ii* 

2.655 

67  33  3i.o* 

17.45 

i383 

7723 

26  Pegasi,               d 

4 

2  37.97* 

3.o33 

84  32  17.3* 

17.53 

1  384 

773i 

29  Pegasi,               Tr2 

4 

3   i9.76* 

2.658 

67  33  21.8* 

17.54 

i385 

7746 

Lacertae, 

5 

5  19.65 

2.3o4 

39  55     0.6 

17.60 

i386 

7749 

21  Cephei,              f 

4 

5  39.53* 

2.071 

82  32   12.9* 

17.60 

i387 

7756 

Gruis,                ft1 

5 

6  33.  93 

3.667 

i32     5  29.4 

17.54 

1  388 

7758 

s4  Cephei, 

5 

6  54.  77 

I  .  I70 

18  23  5o.5 

17.  65 

i388 

7765 

Lacertae, 

5 

7  26.  87 

2.574 

61     i  43.4 

17.53 

1390 

7767 

Tucanee,             a 

3 

8  10.83 

4.194 

161     o  i4.5 

17.72 

i39i 

7773 

43  Aquarii,             & 

ft 

8  54.85* 

3.176 

98  3i  4i.6* 

I7.75 

i392 

7777 

i   Lacertae,. 

5 

9  26.24 

2.6o5 

62  69  47.4 

17.79 

i393 

7778 

23  Cephei,               e 

4i 

9  3i.i3* 

2.197 

33  42  ii.i* 

17.81 

i394 

7788 

3o  Pegasi, 

5 

12  54.79 

3.025 

84  57  46.5 

17.88 

i395 

779° 

47  Aquarii, 

5 

i3  19.93 

3.3i9 

112    20    49.7 

17.89 

i396 

7795 

48  Aquarii,             y 

3 

i3  54-42* 

3.io6 

92     8  28.5* 

17.99 

i397 

7796 

3  1   Pegasi, 

P 

i4    8.29 

2.955 

78  32  62.7 

17.99 

1398 

7800 

2  Lacertae, 

5 

l4    50.22* 

2.465 

44  i3     1.9* 

18.00 

i399 

7808 

Tucanae,             6 

5 

16  35..73 

4.348 

i55  43  41.9 

17-87 

i4oo 

78i4 

62  Aquarii,              TT 

5 

22  i  7  36.99* 

4-3.o68 

89  22  55.4* 

—  18.10 

TABLE  XXX. 


455 


CATALOGUE  OP  1500  STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

c 

d 

a' 

V 

c' 

d! 

i35i 

+9.0490 

-8.9186 

-j-O.2O70 

+8.9925 

—  9.8635 

—  9.8485 

—  I  .2073 

—9-7747 

i352 

8.7553 

8.6234 

0.5254 

—8.2798 

—  9.3228 

+9-43oi 

1.20-78 

9-773? 

i353 

8.8118 

8.673i 

o.5557 

-8.556i 

—  8.54:6 

+9.6522 

I.  2102 

9.760.2 

1  354 

8.9295 

8.  -787-7 

o.3267 

+8.8x69 

—9.8618 

—  9.7965 

I.2II2 

9.76-72 

i355 

8.7386 

8.5966 

0.4690 

+7-94i9 

—  9.7i75 

—  9.1124 

I.2II3 

9.767i 

i356 

8.7524 

8.  6081 

o.4529 

+  8.2IOO 

—  9.7655 

—  9.3674 

I.2I2I 

9.7655 

i357 

8.7882 

8.6437 

0.4241 

+8.46o7 

—9.81-76 

—9.  5825 

I  .2121 

9.7655 

i358 

8.777o 

8.63o8 

0.4328 

+8.4022 

—  9.8o52 

—  9.5358 

I.2I27 

9.7643 

i359 

8.7546 

8.6o48 

o.5i9o 

-8.2i56 

—  9.  395o 

+9.3?28 

I.2l39 

9.7619 

i36o 

8.8o55 

8.655i 

o.5497 

-8.5247 

—  8.8274 

+9.63n 

I  .2l42 

9.7615 

i36i 

9.2i6o 

9.0624 

9.94?6 

+9-  J9°7 

—9.  8364 

-9.8876 

I  .2l52 

9.7694 

i362 

9.o455 

8.8862 

o.2377 

+8.9849 

—  9.85io 

—9.8542 

I  .2I7I 

9.7555 

i363 

9.24o4 

9.o8o8 

9.89i2 

+  9.21-7-7 

—  9.83o5 

—  9.8923 

I  .2I72 

9.7553 

1  364 

8.9i87 

8.7588 

0.3438 

+8.7939 

—9-8536 

—9.  -7903 

I.2I72 

9-7552 

i365 

8.8017 

8.6339 

0.4226 

+8.4937 

-9.8i58 

—  9.6096 

1.2198 

9.7498 

i366 

9.i977 

9.O252 

o.o335 

+  9.1692 

—  9.829i 

—  9.89o5 

I  .22l3 

9-7465 

i367 

8.8474 

8.673o 

o.5627 

—  8.6374 

+7-°792 

+9-7°96 

I  .22I9 

9.7452 

i368 

8.7575 

8.58i9 

o.5i3i 

—8.1488 

—9.4532 

+9.3n3 

I  .2222 

9.7444 

i369 

8.9962 

8.8101 

o.6i7i 

—8.9133 

+9.3i47 

+9.84o2 

I  .2254 

9-737I 

i37o 

9.o448 

8.8579 

0.6354 

—8.9811 

+9.4014 

+9.8597 

1.2256 

9.  7365 

i37i 

8.8n5 

8.6o64 

o.5398 

—8.4994 

—9.0828 

+9.6166 

i.23o9 

9.7236 

I372 

8.757o 

8.5377 

O.492I 

-7.4577 

—  9.6101 

+8.6332 

1.2348 

9.7l32 

i373 

8.8756 

8.6498 

O.5622 

—8.6861 

—6.9031 

+9-7447 

1.2365 

9.7o85 

i374 

9.2793 

9.o534 

9.9582 

+9.  2586 

—  9-79°9 

—9.9136 

1.2365 

9.7084 

i375 

8.7594 

8.5292 

0.4890 

—  7.0209 

—  9.628o 

+8.i97o 

1.2876 

9.7o52 

i376 

8.76o6 

8.53o3 

0.4799 

+  7.6383 

-9.673o 

—  8.8i3i 

i.2377 

9.  7o5i 

i377 

8.7739 

8.5426 

o.5n5 

—  8.i752 

—9.4660 

+9.337i 

i.2379 

9.7044 

i378 

8.93i9 

8.6988 

o.58n 

—8.8008 

+8.9o85 

+9.8o5i 

1.2384 

9.  7o3o 

i379 

9.o948 

8.8589 

O.2520 

+9.o423 

—9.8102 

-9.  8843 

I  .239I 

9.7010 

i38o 

9.H74 

8.88x3 

o.23o6 

+9.0707 

—  9.8072 

—  9.8902 

I  .2392 

9.7008 

i38i 

8.8028 

8.564i 

o.44i6 

+8.4225 

—  9.  -7801 

-9.5572 

1.2898 

9.6989 

i382 

8.8379 

8.588o 

0.4239 

+8.5674 

-9.7988 

—  9.6698 

I  .2425 

9.6904 

i383 

8.7663 

8.5i6i 

0.4783 

+7-7448 

—  9.6800 

—  8.9i89 

I  .2426 

9.6902 

i384 

8.8387 

8.5855 

0.4243 

+8.5683 

—9-7978 

—  9.67o7 

1.2433 

9.6879 

i385 

8.9698 

8.6976 

o.36a5 

+8.8446 

—9.8105 

—  9.828o 

1.2454 

9.6810 

i386 

9.o368 

8.773i 

o.3i55 

+8.9626 

—  9.8040 

—  9.860.4 

1.2458 

9.6799 

i387 

8.8979 

8.63oi 

o.56i6 

—  8.  -7242 

—7.5563 

+9-77°8 

1.2467 

9.6767 

i388 

9.2696 

9.OOO2 

0.0669 

+9.2468 

—  9.7683 

—  9.922I 

i.  2471 

9.6755 

i389 

8.8786 

8.6068 

o.4o83 

+9.6772 

—  9.8021 

—  9.744o 

1.2476 

9.6736 

i39o 

9.o845 

8.8o94 

0.6234 

—  9.0264 

+9.3002 

+9.8880 

1.2483 

9.6710 

i39i 

8.7756 

8.497i 

O.5OO2 

—7-9467 

—  9.5564 

+9.1180 

1.2491 

9.6688 

l392 

8.869o 

8.588o 

o.4i55 

+8.6485 

—9.7964 

—  9.7269 

1.2496 

9.6665 

i393 

9.02-72 

8.7459 

o.33o7 

+8.9472 

—  9-7969 

—  9.8675 

1.2497 

9.6662 

i394 

8.7764 

8.4792 

0.4797 

+7.7i99 

-9.6739 

—8.  8943 

i.253o 

9-6536 

i395 

8.8o9o 

8.5o99 

0.6208 

—  8.389o 

-9.3632 

+9.53i2 

1.2534 

9.6621 

i396 

8.7759 

8.474i 

0.490^ 

-7.3484 

—  9.62OI 

+8.5242 

1.2689 

9.6499 

i397 

8.7846 

8.48i6 

0.4698 

+8.0824 

—  9.7io8 

—9.2498 

i.254i 

9.6490 

i398 

8.933o 

8.6268 

o.SgiS 

+8.7884 

-9.  -7924 

—  9.80-79 

1.2548 

9.  6463 

i39g 

9.1642 

8.8495 

0.6398 

—  9.124o 

+9.3406 

+9.9140 

1.2662 

9.6896 

i4oo 

+8.  779 

-8.4594 

+0.4864 

+6.8120 

—  9.642i 

-7.988o 

—  i  .2574 

—  9.6355 

456 


TABLE  XXX. 


CATALOGUE    OF    1500    STARS. 


No. 

B.A.  C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist., 
Jan.  1,  1850. 

Annual 
Variation. 

h.    m.         s. 

s. 

0            /                // 

" 

l4oi 

7816 

3  Lacertae,            ft 

4* 

22   17  4o.35* 

+  2.343 

38  3i    16.7* 

—  17.92 

I4O2 

7820 

4  Lacertae, 

5 

18  26.49 

2.416 

4i   16  57.9 

18.10 

i4o3 

7828 

Gruis,                 (51 

4 

20  17.06 

3.624 

i  34  i5  35.5 

i8.i5 

i4o4 

783o 

Gruis,                 <5a 

5 

20    46.75 

3.632 

i  34  3o  56.6 

i8.o3 

i4o5 

7832 

55  Aquarii,              £ 

4 

21        6.37* 

3.092 

90  47     8.1* 

18.27 

1406 

7840 

57  Aquarii,              o 

5 

22    42.2O* 

3.i84 

101   26  37.3* 

i8.36 

1407 

784i 

Tucanae,             v 

5 

22    47.98 

4.n8 

i52  45    4.1 

18.16 

t4o8 

7842 

17  Piscis  Aust.,     ft 

4 

22    57.92* 

3.437 

123     6  5o.6* 

18.26 

[409 

7845 

5   Lacertas, 

5 

23     18.85 

2.5o3 

43     3  33.9 

i8.3i 

i4io 

7848 

27  Cephei,               6 

41 

23  36.  60* 

4-2.213 

32  21     5.3* 

i8.3o 

i4" 

7861 

Ursae  Minoris, 

n 

24  So.og* 

—  3.5oi 

4  38  59.3* 

i8.35 

1412 

7855 

7  Lacertae,             a 

4 

25       7.35* 

-r-2.457 

4o  29  i4«4* 

18.35 

i4i3 

7867 

28  Cephei, 

H 

25  3i.26 

o.537 

ii   58  41.8 

i8.36 

i4i4 

7864 

59  Aquarii,              v 

5 

26  28.91 

3.296 

in  28  26.5 

i8.3i 

:4i5 

7868 

62  Aquarii,              TJ 

4 

27  38.77* 

3.087 

90  53  20.3* 

i8.4o 

i4i6 

7886 

Octantis,            ft 

5 

30    21.23 

6.688 

172     9  53.6 

i8.53 

1417 

7896 

3  1  Cephei, 

5 

32     3.90* 

1.486 

17     8     4.8* 

18.66 

i4i8 

7898 

1  8  Piscis  Aust.,      e 

4 

32    21.  OI* 

3.338 

117  49  27.6* 

18.59 

1419 

7902 

3o  Cephei, 

5 

33  20.56* 

2.II4 

27  ii   39.3* 

18.62 

l42O 

79°4 

Gruis,                ft 

3 

33  41.22 

3.626 

i  37  4o     1.8 

18.60 

1421 

7908 

42  Pegasi,              C 

3 

33  58.  89* 

2.990 

79  57     o.3* 

18.68 

l422 

7914 

43  Pegasi,               o 

5 

34  43.24* 

2.809 

61   28  25.8* 

18.67 

1423 

7923 

44  Pegasi,              77 

3 

35  58.57* 

2.8o5 

60  33  43.o* 

18.71 

1424 

7925 

Gruis,                 7} 

5 

36  23.  5o 

3.73i 

i44  17   i8.3 

i8.48 

i425 

7943 

46  Pegasi,              f 

5 

Sg    12.  O2 

2.993 

78  35  38.4 

18.39 

1426 

7945 

47  Pegasi,               A 

41 

39  18.66 

2.883 

67  i3   19.4 

18.  85 

1427 

7946 

Gruis,                 e 

4 

39  28.04 

3.666 

142     6  i4.5 

i8.83 

1428 

7958 

48  Pegasi,               p 

4 

42  46.10* 

2.888 

66  ii   20.8* 

18.91 

1429 

7961 

Cephei, 

5 

43  34.66 

2.443 

34  53  3i.3 

18.96 

i43o 

7966 

22  Piscis  Aust.,      7 

5 

44  10.71 

3.36i 

123  4o  i4-8* 

19.02 

i43i 

7967 

32  Cephei,              i 

4 

44  21.28* 

2.  Il4 

24  35  i5.6* 

i8.85 

i432 

797° 

73  Aquarii,              3, 

4 

44  47.10* 

3.i33 

98    22    34.8* 

19.05 

i433 

7973 

Cephei, 

5 

45  3i.53 

2.3o4 

29     5  59.3 

19.08 

i434 

7980 

76  Aquarii,              d 

3 

46  40.95* 

+  3.i95 

i  06  37     1.4* 

19.06 

i435 

7990 

Cephei, 

51 

47  55.54* 

—  0.006 

7  38  32.o* 

19.  12 

i436 

7992 

24  Piscis  Aust.,      a 

i 

49  21.  o3* 

+  3.335 

120    24    57.1* 

18.96 

i437 

8008 

Gruis,                 £ 

5 

52     0.19 

+  3.6io 

i43  33  22.8 

19.27 

i438 

8o23 

i   Andromedae,      o 

4 

55     1.79* 

+2.745 

48  28  44.3* 

19.28 

i439 

8026 

Cephei, 

5* 

55  25.07 

—  o.i46 

6  27  23.7 

19.  3i 

i44o 

8o3i 

4  Piscium,            /3 

5 

56  14.67* 

+3.o57 

86  59  ii.  3* 

19.28 

i44i 

8o32 

53  Pegasi,              ft 

2 

56  3o.53* 

2.898 

62  43  46.8* 

19.  46 

1  442 

8o34 

54  Pegasi,               a 

2 

57  17.51* 

2.985 

75  36     2.8* 

19.  32 

i443 

8039 

Cephei, 

5 

57  50.89 

2.25l 

23  35  55.2 

I9.35 

1  444 

8o43 

Gruis,                 6 

5 

58  24.79 

3.4i4 

i  34  19  43.8 

19.28 

i445 

8o5i 

55  Pegasi, 

5 

5g  27.03 

3.021 

81   23  58.4 

19.37 

i446 

8o52 

56  Pegasi, 

4* 

59  48.85 

2.916 

65  20  25.o 

i9.37 

i447 

8062 

88  Aquarii,              c* 

44 

23     i   26.57 

3.212 

in   59     6.2 

19.  48 

1  448 

8067 

Gruis,                i 

5 

i   50.78 

3.419 

i36     3  23.8 

19.70 

i449 

8069 

89  Aquarii,              c3 

5 

i   53.85 

3.219 

n3   16     7.2 

19.46 

i45o 

8074 

33  Cephei,               TT 

5 

23     3     8.53* 

+  1.882 

l5    25    22.1* 

—  19.39 

TABLE   XXX. 


457 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b                 c 

d 

a'         \         b' 

c' 

df 

i4oi 

+8.9847 

—  8.6649 

+0.3702 

+8.8782 

-9.784i 

—9.8486 

—  1  .25-72 

—9.6353 

1402 

8.9604 

8.6367 

0.3834 

+8.8363 

-9.7843 

—  9.83i8 

i.258i 

9.6322 

i4o3 

8.9264 

8.5937 

o.5587 

—8.7-702 

—8.2201 

+9.8oi3 

i  .2598 

9.6248 

l4o/! 

8.9288 

8.5936 

0.5590 

—  8.7745 

—  8.l847 

+9.8o37 

i  .2602 

9.6228 

r4o5 

8.7822 

8.4454 

0.4883 

—  6.9i92 

-9.63i7 

+  8.o953 

i  .26o5 

9.6214 

i4o6 

8.7923 

8.4474 

0.5027 

—  8.o898 

—  9.536i 

+  9.2572 

i  .2619 

9.6148 

i4o7 

9.  1229 

8.7776 

0.6169 

—  9.o7i8 

+9.2232 

+9.9086 

1.2619 

9.6144 

Uo8 

8.8607 

8.5i46 

0.5352 

—  8.5982 

—9.1433 

+9.6973 

i  .2621 

9.6i37 

1409 

8.9498 

8.6019 

o.SgSS 

+8.8i35 

—9.7749 

—9.8239 

i  .2622 

9.6122 

i4io 

9.o559 

8.7o64 

+0.3442 

+8.9826 

-9.76i3 

—9.8871 

i  .2626 

9.6110 

i4n 

9.8762 

9.5219 

—0.5535 

+9.8747 

—9.6419 

—  9.9598 

1.2634 

9.6069 

1412 

8.973i 

8.  6160 

+0.3875 

+8.8543 

—9.7692 

—  9.8428 

1.2639 

9.6o45 

i4i3 

9-4688 

9.  1096 

9.7370 

+9.4592 

—  9.683i 

—  9.952^ 

1.2642 

9.6028 

i4i4 

8.81-79 

8.4537 

o.5i58 

—  8.38i5 

—  9.4125 

+9.5264 

i.265o 

9.5986 

i4i5 

8.7877 

8.4174 

0.4884 

—  6.9785 

—9.6312 

+8.i546 

1.2660 

9.5934 

1416 

9.  6552 

9.27o6 

0.8309 

—  9.65i2 

+9.5o3i 

+9.96i8 

1.2681 

9.5812 

1417 

9.3219 

8.9280 

o.i6o5 

+  9.  3022 

—9.6863 

—  9.9475 

1.2694 

9.5733 

i4i8 

8.8447 

8.4492 

o.523o 

—  8.5i38 

—9.3214 

+9.6365 

i.2697 

9.57i9 

1419 

9.  l322 

8.73i2 

0.3243 

+9.o8i3 

—  9.7216 

—9.9173 

I-2704 

9.5672 

1420 

8.9641 

8.56i2 

o.5576 

—8.8328 

—  8.3o75 

+9.83?2 

1.2707 

9.5656 

1421 

8.7993 

8.3948 

0.4748 

+8.o4n 

—9.6914 

—  9.2io5 

I  .2-709 

9.5642 

l422 

8.8493 

8.44o7 

0.4482 

+8.5284 

—9.7442 

—  9.6482 

I.27I^ 

9.56o6 

1423 

8.854i 

8.4384 

0.4472 

+8.5456 

-9.  7436 

—  9.66i7 

I  -2724 

9.5544 

1424 

9  .0282 

8.6101 

O.5722 

-8.9377 

+8.56ii 

+9.88oo 

1.2727 

9.5524 

i425 

8.8o5o 

8.37o8 

0.4739 

+  8.  ion 

—  9.6935 

—  9.2686 

1.2747 

9.5382 

1426 

8.83r7 

8.3968 

0.4589 

+8.4i96 

—9.7272 

—  9.56o4 

1.2747 

9.5376 

1427 

9.0082 

8.5724 

o.5637 

—  8.9o54 

+7.6990 

+9.8698 

1.2748 

9.5368 

1428 

8.8374 

8.38i9 

o.4587 

+8.4435 

—9.7247 

—  9.58o9 

1.2771 

9.5i94 

1429 

9.0419 

8.58i4 

o.3879 

+8.9558 

—9.7132 

—  9.8893 

1.27-76 

9-5i5o 

i43o 

8.8794 

8.4i53 

0.5263 

—8.6233 

—9.2579 

+9.7196 

i  .278o 

9.6117 

i43i 

9.1806 

8.7r54 

0.327I 

+9.i394 

-9.6770 

—9.9346 

I.278l 

9.5io7 

i432 

8.8o48 

8.3369 

o-496i 

—  7.968i 

—9.5824 

+9.i396 

1.2-784 

9.5o83 

i433 

9.n36 

8.64n 

0.3624 

+9.o55o 

—  9.6902 

—  9.9181 

1.2789 

9.5o42 

i434 

8.8198 

8.34oo 

+o.5o46 

—8.2762 

—  9.5i3i 

+9.4337 

1.2-796 

9.4976 

i435 

9.6783 

9.  igoS 

—8.0682 

+9.6744 

—  9.56i5 

—  9.9743 

1.2804 

9.49o4 

i436 

8.8672 

8.3702 

+o.5i97 

—8.5716 

—9.3436 

+9.6834 

1.2813 

9.4820 

i437 

9.o3o7 

8.5i6o 

+0.5563 

—  8.9362 

—8.354i 

+9.886i 

1.2828 

9.4659 

i438 

8.93i9 

8.3963 

+o.4377 

+8.7534 

—9.7078 

—  9.8o38 

1.2845 

9.4467 

i439 

9.7555 

9.2I7I 

—  9.33i6 

+9-7527 

—9.5120 

—9.9798 

1.2847 

9-4442 

i44o 

8.8o75 

8.2632 

+0.4845 

+7.5283 

—  9.65n 

—8.7037 

1.2852 

9.4387 

i44i 

8.8582 

8.3i2i 

o.4597 

+8.5i93 

—9.7084 

—9.6442 

1.2853 

9.437o 

i442 

8.82i3 

8.2695 

o.4739 

+8.2170 

—9.6880 

—  9.3792 

1.2858 

9.43i7 

1  443 

9.2o53 

8.6494 

0.3523 

+9-i674 

—  9.6224 

—  9.9459 

1.2861 

9.4279 

i444 

8.9535 

8.3935 

o.5337 

—8-7979 

—  9.0966 

+9.8285 

1.2864 

9.424i 

i445 

8.8i35 

8.2457 

o.4797 

+  7.9883 

—9.6704 

—  9.i594 

1.2869 

9.4i69 

i446 

8.85o3 

8.2798 

o-464i 

+8.4707 

—  9.7013 

—  9.6o52 

1.2871 

9.4i43 

i44? 

8.8424 

8.2594 

o.5o6i 

—  8.4i57 

—9.  4883 

+9.559o 

1.2879 

9.4027 

i448 

8.9685 

8.3823 

o.5339 

—8.  8258 

—9.0810 

+9.8432 

1.2881 

9.3997 

1449 

8.8466 

8.2601 

o.5o72 

-8.4433 

-9.4765 

+9.5825 

1.2881 

9.3994 

i45o 

+9.3856 

—8.7893 

+0.2743 

+9.3697 

—9.5467 

—  9.9-706 

—  1.2887 

—  9.39o2 

458 


TABLE  XXX. 


CATALOGUE  OF  1500  STARS. 


No. 

B.A.C. 

Constellation. 

Mag. 

Right  Ascension, 
Jan.  1,  1850. 

Annual 
Variation. 

North  Polar  Dist.  ,     Annual 
Jan.  1,  1850        Variation. 

h.      m.       s. 

s. 

0               /                    "//        ~ 

i45i 

8082 

7  Andromedae, 

5 

23      5   41.49* 

+  2.724 

4i   24  44.8* 

-19.  58 

1452 

8o85 

90  Aquarii,              § 

5 

6  33.i4* 

3.n4 

96  5i  24.1* 

19.35 

i453 

8093 

Tucanse, 

5 

7  4?.  32 

3.423 

i  52  47  61.9 

22.39 

i454 

8098 

Tucanae,              7 

4 

8  38.54 

3.554 

149     3  28.2 

19.55 

i455 

8io5 

6  Piscium,             7 

41 

9  23.36* 

3.  no 

87  32  ii.  i* 

19.61 

i456 

8109 

93  Aquarii,              i/>2 

5 

10     6.54 

3.128 

IOO       O        2.8* 

19.56 

i457 

8n3 

Sculptoris,          7 

5 

10  42.66* 

3.260 

123     20     54.9* 

19.  52 

i458 

8u4 

8  Andromedae,                5 

10    48.22* 

2.756 

4i   48    i3.4* 

19.61 

i45g 

8116 

95  Aquarii,              tp3 

5 

II        9.27* 

3.128 

100   25  47.9* 

19.62 

i46o 

8i3i 

62  Pegasi,               T 

5 

i3  i3.i8* 

2.961 

67    4  47.7* 

19.65 

i46i 

8i44 

98  Aquarii,              bl       5 

i5     5.29 

3.i63 

no   55     5.3 

19.60 

1462  8160 

68  Pegasi,               v 

5 

17  53.91* 

2.986 

67  25   i5.4* 

19.  79 

i463 

8161 

99  Aquarii,              V* 

5 

18     9,77 

3.i63 

in   27  46.2 

19.70 

i464 

8162 

4  Cassiopcas, 

5 

18  ii.  95* 

2.627 

28  32   24.4* 

19.74 

i465 

8i77 

10  Piscium,             6     \  5 

20  21.  61* 

3.o42 

84  26  4o.o* 

19.73 

i486 

8180 

Cephei, 

5 

20  57.42* 

2.463 

2O    27     54.1* 

19.77 

i467 

8182 

7o  Pegasi,               q 

5 

21     34.24 

3.028 

78     3   57.3 

19.83 

i468 

8188 

Cassiopeae, 

5 

23     7.67* 

2.747 

32   16  4o.8* 

19.82 

i-469 

8201 

Sculptoris,         /? 

5 

24  54.82 

3.238 

128    38   5o.6 

19.72 

i4yo 

8202 

101  Aquarii,              Z»* 

5 

25  25.38 

3.i52 

in   44   33.5 

19.87 

i47i 

8210 

Phoenicis,           c 

5 

26  59.21 

3.247 

i33  26   38.6 

19.70 

1472 

82i3 

Urss3  Minoris, 

51 

27  48.99* 

0.073 

3   3i    ii.  8* 

19.88 

i473 

8224 

1  6  Andromedae,      h 

4* 

3o  i4.43* 

2.912 

44  21    i5.3* 

19.  5o 

i474 

8229 

1  7  Andromedae,       c 

4 

3o  47.57* 

2.918 

47   33   42.7* 

19.92 

i475 

8232 

1  02  Aquarii,              ul 

5 

32     o.i3 

3.119 

io5     3     4.1 

19.89 

i476 

8233 

17  Piscium,             i 

& 

32   i4-34* 

3.  112 

85   ii    10.7* 

19.  47 

i477 

8237 

19  Andromeda?,      K 

41 

33     1.99* 

2.927 

46  29  44.8* 

19.94 

i478 

8238 

35  Cephei,              7 

3 

33   14.02* 

2.39I 

i  3    12    i  6.  6* 

20.09 

i479 

8240 

io3  Aquarii,              A1 

5 

33  47.62 

3.123 

1  08   5  1    1  8.  8 

19.  87 

i48o 

8242 

io4  Aquarii,              A2 

5 

33  58.44 

3.126 

108   38   5o.5 

20.00 

i48i 

8243 

1  8  Piscium,            A 

5 

34  23.55* 

3.o64 

89     2  42.4* 

19.80 

1482 

8255 

1  06  Aquarii,              il 

5 

36  25.17 

3.123 

109     6   3i.3 

19.98 

i483 

8266 

78  Pegasi, 

5 

36  27.33 

3.io6 

61   28     5.6 

19,96 

i484 

8261 

20  Andromedae,      1/1 

5 

38  36.99* 

2.948 

44  24.  44.2* 

19.96 

i485 

8268 

5  Cassiopeae,         r 

5 

39  44-77* 

2.891 

32     II        O.2* 

20.02 

i486 

8273 

Cephei, 

5 

4o  46.33 

2.8i3 

23     i    34.8 

19.99 

i487 

8275 

Sculptoris,          d 

5 

4i     6.39* 

3.i4i 

118   57   33.4* 

19.89 

i488 

8290 

Octantis,             y1 

5 

43     5.7o 

3.7i5 

i75  5i     9.9 

19.98 

r489 

83i4 

Cephei, 

5 

47  SS.go 

2.852 

16  a5  24.9 

20.  iL 

1490 

83i9 

Octantis,             7* 

5 

49     8.23 

3.474 

i73     o  18.4 

19.  79 

1491 

8323 

Tucanae,             7} 

5 

49  39.75 

3.i7o 

i55     7  5o.5 

20.  II 

1492 

8328 

2-7  Piscium, 

5 

5o  59.58* 

3.072 

94  23   18.0* 

19.92 

i493 

833i 

28  Piscium,             u 

41 

5i   36.63* 

3.080 

83  58     1.8* 

19.96 

i4g4 

8334 

Tucanse,             e 

5 

52     4.39 

3.179 

i56  24  43.i 

19.94 

i495 

8344 

Cassiopeae, 

5 

53  59.6o* 

3*.  009 

29  36  44-6* 

20.  04 

1496 

8346 

29  Piscium, 

5 

54    8.17* 

3.o75 

93  5i  44.5* 

20.07 

i497 

8349 

3o  Piscium, 

41 

54  i5.96* 

3.082 

96   5o  52.o* 

20.  03 

1498 

8358 

2  Ceti, 

4 

56     3.12* 

3.082 

i  08   10  i  4.  9* 

20.07 

1499 

8366 

Cassiopese, 

5 

5?  23.oo* 

3.o44 

29  3i    1-7.2* 

20.  o5 

1600 

8368 

33  Piscium, 

5     |23  57  39.39* 

+  3.076 

96  32  48.  7* 

—  2O.  10 

TABLE    XXX. 


459 


CATALOGUE    OF    1500    STARS. 


No. 

Logarithms  of 

Logarithms  of 

a 

b 

e                  d 

of 

V 

c' 

d' 

i45i 

+  8.9911 

—  8.374o 

+o.4337 

+8.8661 

—9.6668 

—  9.862-7 

-   .2899 

—  9.37o6 

1462 

8.816! 

8.1908 

0.4926 

—  7.8921 

—9.6066 

+  9.0660 

.29o3 

9.3638 

i453 

9.i525 

8.5i77 

0.5633 

—  9.  1016 

+  7-3979 

+  9.937-7 

.2909 

9.3538 

i454 

9.  1018 

8.4696 

0.6622 

—  9.o35i 

—  8.6119 

+  9.9223 

.29I2 

9.3468 

i455 

8.8i36 

8.i648 

o.4854 

+7-447I 

9.  6464 

—8.6228 

.29i5 

9.3406 

i456 

8.8202 

8.i65o 

0.4944 

—  8.0699 

9.6910 

+  9.2294 

.29I9 

9.3344 

i457 

8.8919 

8.2312 

o.5i3o 

—8.6321 

9.3895 

+  9.7300 

.292I 

9.3292 

i458 

8.9900 

8.3285 

o.4397 

+8.8624 

9.6614 

—9.8623 

.292I 

9.3284 

r459 

8.8212 

8.i565 

0.4945 

—  8  .o79o 

9.5894 

+9.2478 

.2923 

9.  3253 

r46o 

8.85o5 

8.i665 

o.47o7 

+8.44io 

9.6800 

—9.58i3 

.293i 

9.3069 

i46i 

8.845i 

8.1429 

o.5on 

—  8.3978 

9.6262 

+9.5443 

.2938 

9.2894 

1462 

8.8612 

8.1202 

o.4726 

+8.4355 

9.6-728 

—  9.5769 

.2949 

9.2617 

i463 

8.8478 

8.n4i 

0.6006 

—8.4112 

9.5283 

+9.6661 

.2949 

9.2690 

i464 

9-i374 

8.4o33 

0.4191 

+9.0812 

9-5643 

—9.9366 

.295o 

9.268-7 

i465 

8.8194 

8.o6i7 

o.484o 

+7.8o54 

9.6619 

—  8.9-794 

.2957 

9.2358 

i466 

9-274o 

8.6096 

0.3916 

+9.2457 

9.4921 

—9.9664 

.2959 

9.2293 

i467 

8.8273 

8.o558 

0.4806 

+8.1428 

9.6619 

—  9.3o94 

.2961 

9.2224 

i468 

9.o9o7 

8.3oio 

0.4364 

+9.oi78 

9.6648 

—  9.92i5 

.2966 

9.2047 

i469 

8.9262 

8.ii45 

0.609-7 

-8.7217 

9.3869 

+9.7904 

.297I 

9.1833 

i47o 

8.85io 

8.0329 

0.4984 

—8.4197 

9.6408 

+9.5638 

.2973 

9.i77o 

i47i 

8.9584 

8.  I2OO 

O.5l22 

—  8.7958 

9.3353 

+9.8329 

•2977 

9  .  16-70 

i472 

o.o3i5 

9.1823 

8.3892 

+  o.o3o6 

9.2214 

—9-9949 

•2979 

9.i466 

i473 

8.9757 

8.o9i7 

o.46i4 

+  8.83oo 

9-5977 

—  9.85o7 

1.2986 

9.1123 

i474 

8.9623 

8  .0600 

o.4647 

+  8.78i5 

9.6o86 

—  9.8256 

1.2987 

9.  io4i 

i475 

8.8358 

•7.9249 

0.4934 

—  8.25o3 

9.5883 

+9.4ii2 

I  .299O 

9.o859 

i476 

8.8223 

7-9°77 

0.4853 

+7.746i 

9.6469 

—  8.92O7 

I  .2990 

9.0822 

i477. 

8.9604 

8.o33i 

0.4656 

+8.7982 

9.5973 

—  9.8348 

.2992 

9.o697 

i478 

9.4622 

8.53i6 

o.38i4 

+9.45o5 

9.3274 

—  9.9854 

.2993 

9.o664 

i479 

8.845o 

7.9061 

0.4945 

—8.3545 

9.5726 

+9.5o66 

.2994 

9.o573 

i48o 

8.8445 

7.9016 

0.4944 

—  8.3493 

9.574o 

+9.5o2o 

•  2994 

9.o543 

i48i 

8.82i3 

7.87I2 

0.4869 

+7.o434 

9.6393 

—  8.2194 

.2996 

9.o472 

1482 

8.8462 

7.8601 

0.4939 

—  8.36i3 

9.5758 

+9.5i27 

.2999 

9.ou6 

i483 

8.8778 

7.8911 

o.4766 

+  8.6669 

9.6338 

—  9.6768 

.2999 

9.oio9 

i484 

8.977o 

7.9482 

0.4689 

+8.83o9 

9.5686 

—  9.852o 

.  3oo3 

8.9692 

i486 

9.0968 

8.o433 

0.4699 

+  9.0233 

9-4854 

—  9.9269 

.3oo5 

8.9458 

i486 

9-23oo 

8.i548 

0.4482 

+9.1-940 

9.3924 

—  9.9624 

.3oo7 

8.0232 

i487 

8.88o4 

7-7975 

0.4968 

—8.5655 

—  9.53i2 

+9.6835 

.3oo7 

8.9i56 

i488 

9>7278 

8.5965 

0.6862 

—9.  -7246 

+8.2810 

+9.9964 

.3oio 

8.8674 

i488 

9.37i9 

8.io56 

o.45n 

+9.3538 

—  9.2420 

—9.9813 

.3oi6 

8.733i 

1490 

9.7378 

8.4i39 

0.5547 

~9.7346 

7-77°9 

+9.9963 

.3oi7 

8.6756 

i49i 

9-T997 

7.8642 

o.5o53 

—  9.i574 

9.1611 

+9-9573 

.3*1* 

8.654i 

1492 

8.8249 

7.4194 

0.4878 

—  7.7086 

9.  6332 

+8.8834 

.3oi9 

8.5942 

i4g3 

8.8260 

7.3897 

0.4865 

+7.8477 

9.6389 

—  9.0213 

,3oi9 

8.5634 

1494 

9.2214 

7.7606 

o".  6020 

—  9.i835 

9.1623 

+9.9619 

.3O20 

8.5388 

i495 

9.1299 

7-5487 

0.4784 

+9.0691 

9.375i 

—9.9391 

.3021 

8.4i86 

1496 

8.8248 

7.232-7 

0.4876 

—7.6532 

9.6348 

+8.8283 

.3021 

8.4o78 

1497 

8.8269 

7.2260 

o.4878 

—7.9033 

9.63i3 

+9.o763 

.3021 

8.398o 

1498 

8.846i 

7.0822 

0.4883 

—  8.34oo 

9.6096 

+9.4939 

.3022 

8.236o 

1499 

9.i3i3 

7.1898 

0.4834 

+9.0709 

9.3499 

—  9.9396 

.3022 

8.0686 

i5oo 

+8.8267 

—6.836o 

+o.4875 

-7.8837 

—9.6334 

+9.o57o 

—     .3022 

8.0092 

4GO 


TABLE   XXXI. 


Secular  Variation  of  the  Annual  Precession  in  Right  Ascension. 


Star. 

Sec.  Var. 

Star. 

Sec.  Var. 

Star. 

Sec.  Var. 

Star,  j       Sec.  Var. 

Star. 

Sec.  Var. 

s. 

s 

s. 

s. 

s. 

2 

+  o.o488 

213 

+  0.0729 

63i 

—  O.  I2O3 

827 

+  o.54i3 

1235 

—  o.  1614 

5 

—    O.22I2 

219 

-{-0.1047 

632 

—  o.o38o 

829 

4-  0.9674 

I25o 

—    O.O962 

9 

—  0.0587 

225 

4~o.o4o4 

64o 

—  o.o693 

83i 

4-  0.4714 

1255 

—  o.o352 

IO 

—  0.0709 

228 

+o.o355 

645 

—  o.o36i 

838 

+  0.0435 

1271 

—  29.  3200 

12 

—  0.0951 

23l 

4-o.o352 

648 

—  0.0725 

844 

4-  0.042  i 

1272 

—  o.o6o3 

16 

-f-  0.0467 

235 

4-0.0376 

653 

—  0.1161 

845 

4-  0.0424 

1273 

—  0.1670 

18 

4-  0.0673 

272 

4~o.o4o5 

654 

—  o.  1470 

848 

4-  o.o52i 

1284 

—  o.o4n 

I9 

—  o.o475 

277 

4-o.  0703 

656 

—  0.0670 

85o 

+  0.0372 

1286 

—    O.I2O5 

20 

—  o.o475 

288 

+0.0977 

662 

—  o.o9i7 

852 

4-  0.0498 

I290 

—  0.1171 

21 

—  0.0469 

289 

+0.0421 

664 

—o.o386 

853 

+    O.I202 

I294 

—  o.  i449 

23 

-f-  o.o468 

293 

4-o.  0372 

668 

—0.0653 

855 

4-  0.0860 

I298 

—  0.3856 

27 

4-  0.0529 

3oo 

4-0.2235 

671 

—  0.0848 

856 

4-  0.0860 

1299 

—  22.  692O 

28 

—  o.o384 

3o3 

4-o.o668 

675 

—  o  ,o39o 

857 

4-  0.4195 

i3o8 

—    0.0746 

3i 

+  0.0370 

33i 

+o.o833 

692 

—  o.o579 

858 

4~  0.1099 

i3io 

—  o.363o 

38 

4-  o.o58i 

4o5 

—  o.o864 

694 

—  o.  n59 

859 

4-  o.o455 

i3i5 

—  o.5i36 

39 

4-  0.0439 

4n 

—  1.4765 

698 

+o.o4i8 

874 

+  0.0387 

i3i6 

—  o.3o64 

43 

4-  o.o4i2 

422 

—o.o366 

7o3 

—  o.o38i 

878 

4-  0.0376 

1328 

—  o.  1722 

44 

4-  0.0686 

424 

—  o.o434 

7o5 

+o.o532 

879 

4-  o.o36i 

i334 

—    O.  1221 

46 

4-    1.2222 

425 

—  o.  io54 

710 

—  o.o458 

880 

4~  0.1009 

i337 

—  o.o658 

5i 

4-  o.o36o 

447 

—  o.452o 

711 

+  o.ii54 

882 

+  o.o436 

1  342 

—  o.o368 

53 

4-  o.o4o8 

452 

—0.0375 

7i4 

4-o.o6oi 

889 

4-  o.o4o3 

1  346 

—  0.2683 

5y 

4-  o.o565 

463 

—  o.o366 

7i5 

+o.o356 

892 

4-  0.7265 

i36i 

—  o.o354 

GO 

4-11.4276 

4?i 

—  0.0812 

718 

4-o.o385 

893 

4-   o.o46i 

i363 

—  0.0426 

64 

4-  0.0396 

495 

—  o.  3909 

719 

—  o.i3i9 

894 

4-  o.o535 

i369 

—  0.0682 

65 

4-  0.1173 

498 

—  o.o64i 

720 

+o.o5oo 

895 

4-  o.i385 

1370 

—  0.0837 

66 

4-  0.0748 

5n 

—  o.o423 

721 

—  o.o448 

896 

4-  0.0735 

i373 

—  o.o356 

73 

4-  0.0399 

5l2 

—  o.u85 

727 

+  O.  I732 

897 

4-  o.o438 

i374 

—  o.o389 

74 

+  0.1390 

521 

—  0.0392 

728 

+0.0642 

9o3 

4-  o.o435 

i378 

—  0.0476 

75 

4-  0.0426 

523 

—  0.0619 

732 

+o.o558 

9o5 

+  o.o379 

i387 

—  o.o38i 

81 

+  0.0462 

525 

—  o.o5o2 

735 

+o.o652 

907 

+  0.0724 

i39o 

—  0.0880 

86 

4-  o.o5o8 

529 

—0.0770 

736 

+o.o65i 

910 

4-  0.0729 

l399 

—  o.u49 

92 

4-  0.0968 

532 

—  o.io4i 

74o 

+0.0389 

9i3 

4-  o.  iu5 

i4o3 

—  o.o4o8 

IOI 

4-  0.1606 

533 

—0.1458 

743 

+o.o5i6 

924 

4-  0.0357 

i4o4 

—  o.o4n 

io3 

4-  o.i83o 

534 

—0.0799 

744 

4-O.II23 

929 

4-  0.1913 

i4o7 

—  o.o945 

1  08 

+  o.o374 

536 

—0.1623 

748 

—  o.o582 

934 

4-  0.1244 

i4n 

—    1.1224 

117 

4-  0.0664 

537 

—  o.  1008 

75o 

+0.0961 

938 

+  o.o858 

i4i3 

—  o.o879 

118 

4-  0.1288 

555 

—  0.2169 

75i 

+  o.o383 

95i 

4~    O.2Ol5 

i4i6 

—  0.6890 

127 

4-  0.0491 

557 

—  o.o459 

754 

4-o.o395 

967 

+  0.0789 

l42O 

—  o.o456 

i37 

4-  0.0660 

558 

—0.1376 

757 

4-0.0961 

972 

4-  o.o38i 

1424 

—  o.o599 

i43 

+  o.o568 

56o 

—0.0358 

758 

4-o.o628 

975 

4-  0.0369 

l427 

—  o.o54o 

i5o 

4-  o.o579 

562 

—o.o448 

760 

4-o.74i4 

977 

+  o.33i8 

i435 

—  0.2170 

i5i 

-f-  0.0675 

563 

—  0.1317 

762 

+o.o579 

981 

4-   0.1234 

i437 

—  o.o556 

i58 

4-  0.0482 

566 

—  o.i35o 

766 

+  0.23l4 

997 

4-  0.2495 

i439 

—  0.2923 

1  60 

4-  0.0395 

567 

—  o.o55i 

768 

+  O.232I 

998 

4-   o.n5i 

i444 

—  0.0378 

162 

4-  0.0708 

568 

—  o.  io49 

771 

+  o.i328 

1006 

4-  0.0923 

i448 

—  o.o4oo 

166 

4-  o.nos 

572 

—  0.0928 

775 

4-o.o45o 

1009 

4-  o.o38i 

i453 

—  0.0811 

171 

+  o.2783 

574 

—  o.o632 

783 

+o.o355 

ioi3 

4-  o.o54i 

i454 

—  0.0671 

172 

-j-  0.0470 

575 

—  0.0627 

786 

4-o.uo5 

ioi4 

4-  0.0922 

i464 

+  0.0359 

176 

4-  0.0762 

58i 

—  0.0372 

794 

+  1.3966 

1018 

4-   o.o452 

i466 

4-  o.o455 

177 

+  0.0717 

586 

—  o.8i53 

801 

4-o.o567 

io45 

4-     0.2928 

l472 

—  0.4747 

178 

-f-  0.0599 

589 

—  0.  IO52 

8o5 

4-o.o526 

io5i 

+   o.o543 

i478 

+   o.o7o3 

180 

+  0.1967 

592 

—  o.  1730 

807 

+o.o36o 

1082 

4-2i.  i44i 

i485 

+  o.o4o4 

181 

+    0.0425 

593 

—0.0578 

808 

+o.o37i 

n43 

—  0.6157 

i486 

4~  0.0669 

i83 

4-  0.042  i 

6o5 

—  O.  IIOO 

8i4 

+o.o45o 

1149 

—  0.0371 

i486 

—  0.3599 

188 

4-  o.o4o4 

608 

—0.0839 

817 

4~o.o369 

1162 

—  o.o4i8 

i489 

+  o.o843 

189 

4-  0.0886 

610 

—  o.o652 

818 

4-o.o4o9 

u75 

—  o.o459 

i49o 

—  0.3254 

190 

-f-  0.1602 

622 

—  I  .7282 

820 

4-o.286o 

n77 

—  0.0587 

i49i 

—   o.o7o4 

i93 

4-  o.ion 

623 

—0.1180 

821 

+o.o424 

1209 

—  o.o359 

i494 

—  o.o735 

2IO 

4-  o.o832 

624 

—  o.o4o4 

822 

4-0.0820 

1210 

—  0.0600 

i495 

+  o.o5oi 

2T2 

4-  o.5ri2 

63o 

—  0.9968 

824 

4-o.o358 

1222 

—  o.o4oo 

1499 

4-  o.o5i7 

TABLE    XXXII. 


461 


Secular  Variation  of  the  Annual  Precession  in  North  Polar  Distance. 


Star.)   Sec.Var. 

Star. 

Sec.  Var. 

Star.    Sec.  Var. 

Star.     Sec.  Var. 

Star. 

Sec.  Var. 

Star. 

Sec.  Var. 

60 

4-0.  7i3 

297 

+  0-479 

4i3  +0.621 

592 

+o.5n 

1028 

—  o.58o 

u33 

—  0.457 

160 

o.43o 

298 

0.587 

4i5 

o.5oi 

622 

+o.?I3 

1029 

o.58i 

1  1  34 

o.58o 

i5i 

0.44? 

3oo 

1.367 

4i8 

0.632 

63o 

+0.544 

1032 

0.683 

u36 

o.648 

1  66 

0.549 

3o5 

0.483 

421 

0.533 

827 

—  o.5o8 

io33 

0.443 

n37 

0.671 

172 

0.462 

3o8 

o.579 

422 

o.9o7 

829 

—0.646 

io36 

o.659 

n38 

0.817 

176 

0.529 

3io 

o.444 

423 

0.486 

83i 

—  o.5n 

io38 

0.664 

u4o 

o.SSg 

177 

0.523 

3l2 

0.625 

424 

o.938 

857 

—  0.622 

io4o 

o.55i 

n4i 

0.646 

178 

o.5o3 

3i3 

0.553 

425 

1.273 

892 

+0.496 

1044 

—  o.6o4 

1142 

—0.497 

181 

o.468 

3i7 

0.592 

427 

0.469 

894 

—0.443 

io45 

+  o.9i3 

n43 

+2.807 

i83 

0.481 

322 

o.45o 

428 

o.45o 

895 

0.577 

1046 

—  o.484 

n44 

—0.645 

188 

0.493 

324 

0.542 

429 

0.611 

896 

0.490 

io5o 

O.527 

u46 

0.622 

189 

o.6o3 

325 

0.432 

43i 

o.568 

897 

0.441 

io5i 

0.883 

n49 

1.022 

190 

0.725 

326 

0.461 

433 

0.747 

903 

0.449 

io55 

0.438 

n5i 

0.855 

191 

o.44o 

32y 

o.569 

443 

o.5o5 

9o5 

0.438 

io58 

o.Sio 

u53 

0.856 

102 

0.476 

328 

o.5i6 

447 

1.852 

9i3 

0.607 

io5g 

o.48o 

u54 

0.474 

I93 

0.636 

329 

o.45o 

449 

0.582 

9i4 

o.45i 

1  06  1 

0.718 

n55 

o.54o 

210 

o.63i 

33i 

i.i43 

45o 

0.538 

92O 

0.462 

1062 

O.525 

n56 

o.8o4 

21  I 

0.454 

333 

0.438 

452 

o.743 

924 

0.476 

io63 

0.709 

1162 

o.894 

212 

1.160 

334 

o.46i 

454 

o.43o 

934 

0.696 

io64 

0.666 

u65 

0.519 

216 

0.489 

335 

o.56i 

455 

0.483 

935 

0.455 

io65 

0.524 

1166 

o.532 

218 

0.475 

336 

0.441 

456 

o.64o 

937 

0.432 

1066 

0.774 

n67 

o.5i8 

225 

0.553 

34o 

o.475 

458 

0.585 

938 

0.629 

io67 

0.548 

u7i 

o.Sio 

226 

o.44i 

34i 

o.476 

462 

o.48  1 

942 

o.434 

io7o 

0.665 

11  79 

0.543 

228 

0.542 

345 

0.439 

464 

0.685 

947 

0.465 

I07I 

0.585 

1182 

o.Sog 

p3i 

o.557 

346 

o.475 

465 

0.499 

948 

o.45o 

IO72 

0.526 

n84 

o.574 

233 

o.5i8 

347 

o.5i7 

467 

o.559 

952 

o.444 

io73 

o.585 

n85 

o.53i 

235 

o.574 

348 

0.556 

470 

0.577 

954 

o.486 

io75 

0.620 

1186 

o.Sgi 

240 

0.477 

349 

0.435 

4?i 

o.874 

956 

o.436 

io78 

0.496 

1187 

o.45o 

241 

o.44i 

352 

0.438 

472 

o.5i6 

958 

0.526 

1081 

0.471 

1189 

0.576 

246 

0.449 

355 

o.6o3 

475 

0.448 

960 

0.435 

1082 

:5.545 

1190 

0.583 

248 

o.45o 

356 

0.534 

476 

o.529 

961 

0.491 

io83 

0.849 

1  191 

o.5o3 

25o 

0.453 

359 

o.6o4 

48o 

0.524 

962 

0.439 

io84 

0.600 

1192 

o.456 

25i 

0.469 

36o 

o.78o 

48  1 

0.466 

964 

o.44i 

1086 

0.488 

1193 

o.5i5 

252      0.445 

36i 

o.73o 

482 

o.5o3 

965 

o.458 

io87 

O.52I 

1  196 

0.490 

254 

0.460 

364 

o.548 

488 

o.5o3 

967 

0.683 

1089 

o.43o 

1201 

0.601 

255 

o.45i 

367 

0.519 

49o 

0.485 

968 

0.468 

1090 

0.608 

1202 

0.602 

256      o.45i 

369 

0.472 

491 

o.497 

969 

0.442 

1092 

0.548 

1204 

0.483 

258 

0.453 

37i 

0.717 

496 

1.299 

975 

0.576 

1093 

0.592 

I2O5 

0.577 

261 

0.460 

374 

o.64i 

499 

0.483 

977 

i  .  157 

1094 

0.437 

1209 

o.673 

262 

o.44o 

375 

o.648 

5n 

o.58o 

978 

0.472 

io97 

o.56i 

1216 

o.448 

266 

o.45o 

376 

o.595 

5l2 

o.773 

984 

0.458 

1098 

o.533 

1217 

0.495 

269 

o.45o 

38o 

o.48i 

5i4 

o.448 

986 

0.470 

1099 

o.48i 

1218 

o.436 

270 

o.486 

382 

0.517 

5i6 

0.457 

989 

0.457 

iio3 

0.460 

1225 

o.459 

272 

0.674 

383 

0.532 

523 

o.6o5 

994 

0.485 

uo5 

0.433 

1235 

0.930 

277 

0.807 

384 

o.5i9 

525 

o.56i 

996 

o.5i7 

II07 

o.438 

1238 

o.544 

278 

0.442 

385 

0.772 

526 

o.5o6 

997 

1.117 

no9 

o.84i 

1242 

0.480 

279 

0.449 

386 

o.499 

529 

0.611 

IOOO 

0.455 

IIIO 

0.476 

1249 

o.475 

280 

0.439 

388 

0.786 

532 

0.653 

IOOI 

0.471 

IIII 

0.681 

I25o 

o.74i 

281 

0.467 

389 

0.966 

537 

0.625 

1006 

0.817 

III2 

0.558 

I25l 

—0.488 

282 

0.661 

39i 

0.497 

557 

0.453 

1007 

0.526 

in5 

0.562 

1271 

+6.481 

284 

0.432 

392 

0.808 

558 

o.599 

1008 

0.498 

1116 

0.439 

1272 

-0.585 

285 

o.SSg 

394 

o.558 

562 

o.439 

IOII 

0.443 

1  1  17 

0.808 

1284 

—0.497 

286 

o.468 

395 

0.529 

563 

0.568 

101^ 

0.709 

1118 

o.554 

1286 

-0.654 

287 

0.563 

396 

o.773 

566 

0.565 

I0l2 

o.848 

1120 

o.65o 

1290 

—  o.637 

289 

0.737 

399 

o.673 

567 

o.447 

ioi5 

0.469 

1122 

0.832 

129^ 

—0.663 

290 

o.597 

4oo 

o.528 

568 

O.522 

1018 

0.700 

1122 

0.523 

1299 

+4.64i 

291 

o.582 

4o^ 

o.5i8 

5?2 

0.493 

1022 

0.538 

1126 

—  o.593 

i3o8 

—0.526 

293 

0.724 

4o5 

I  .5l2 

575 

o.44o 

IO2^ 

0.454 

II27 

+  0.653 

i3io 

—  o.83o 

29^ 

o.  5oo 

4o8 

0.470 

586 

0.895 

IO2<! 

0.556 

1128 

+  0.653 

i334 

—0.490 

296 

+0.655 

4n 

+4.45o 

58^ 

+0.452 

1025 

—  o.557 

II29 

—  o.559 

462  TABLE    XXXIII. — ELEMENTS   OF   THE    PLANETARY  SYSTEM. 


Mercury. 
Venus... 
Earth . . . 
Mars.. 


Asteroids. 

Jupiter . 

Saturn 

Uranus 

Neptune 


Name. 


Distance  from  the  Sun. 


Mean. 


Greatest.  Least. 


Eccentricity. 


Sidereal  Revolution. 


0.8870984 
0.7233317 
I .OOOOOOO 
I .623691 

6.202767 
9.53885o 
19. 18239 
3o. 03627 


0.4666927 
0.7282636 
i .0167761 
i .6667796 

5.453663 
10.073278 
20.07630 
30.29816 


,3076041 
,7183998 
,9832249  o 
,3816026 


,961871 
,  004422 
,28848 
•77438 


0.2066178 
0.0068 i 83 
.0167761 
0.0932628 


o.o482235 
0.0660266 
o. 0466006 
0.0087193  6 


Synodical 
Revolu- 
tion. 


Days.          |     Days. 
87.96928241116.877 
224.7007764  683.920 

365.2563744 
686.9794661  779.936 


4332.5848o32 
10769.2197106 
3o686. 8206666 

0126.722 


398.867 
378.090 
369.656 
367.488 


Name. 


Longitude  of  the 
Perihelion. 


Annual 
Variation. 


Longitude  of 
Ascending  Node. 


Annual     Inclination  of 
Variation.          Orbit. 


Annual 
Variation. 


Mean  daily 
Motion. 


Com 
pres 
sion. 


Mercury. 
Venus  . . 
Earth... 
Mars  .. 


Asteroids. 

Jupiter 

Saturn 

Jranus 

S"eptune. . . 


74  67  27.0 

124  i4  26.6 

loo  II  27.0 

333  6  38.4 

ii  45  32.8 

89  54  4i-2 

168  5  24 

47  17  58.o 


+  5.8i 
-  3.24 
+  11.24 
+  16.46 

+  6.65 
+  19. 3i 

+  2.28 


46  23  55.0 

76  ii  29.8 

48  16  18.0 

98  48  37.8 

112  16  34.2 

73  8  47-8 

i3o  10  12.3 


—  10.07 
— 20.60 

— 26.22 


-16.90 
-19.64 
-36. 06 


7  o   i3.3 

3  23   3i.4 

i  5i     6.7 

1  18  42.4 

2  29    29.9 

0  46   29.2 

1  46  69.0 


+0.18 
+0.07 


0.23 

— o.  16 
+o.o3 


246  32.6 

96  7.8 

69  8.3 

3i  26.7 

4  59.3 

2        0.6 

42.4 

21.6 


Name. 


Sun 

Mercury 
Venus.. 
Earth . . 

Mars 

Asteroids. 
Jupiter . 
Saturn  . 
Uranus  _ 
Neptune 


Time  of 
Rotation. 


Diameter. 


Apparent.  |  In  Miles. 


h.   m. 

607  48 


5  28 

21  21 


23  56  4 

24  37  22 

9  55  26 
10  29  17 


1923.64888 
6.69 

17. 10 


5.8 

38.4 

17.1 

4.i 

2.4 


,646 
3,089 
7,896 
7,926 
4,070 


92,164 
76,070 
36,2i6 
33,6io 


Volume. 


0.0696 
0.9960 
I .0000 
o.i364 

1491-0 
772.0 

86.5 
76.6 


Mass. 


35493G 
0.0729 
0.9101 
I .0000 
0.1324 

338. 718 

101.364 

14.261 

18.900 


Density. 

0.260 
I  .225 
o.  908 
1 .000 

0.972 

0.227 
o.i3i 
0.167 

0.321 


Light  at 


Perihelion.  Aphelion 


10.58 
1.94 
i.o34 
0.624 

o . o4o8 

0.0123 

0.0027 

O.OOI I 


4.59 
1.91 
0.967 
o.36o 

o.o336 
0.0099 
0.0026 

O.OOI I 


Gravity. 

28.36 
0.48 
0.90 

I  .00 

0.49 

2.45 
1 .09 
0.76 
1.36 


Bodies 

fall  in 

one  Sec, 


456.6 

7-7 
i4-5 
16.1 

7-9 

39.4 
17.6 

12.3 


The  preceding  elements  of  Neptune  are  for  the  beginning  of  1854;  the  others  are  for  the  be- 
ginning of  1840. 


TABLE    XXXIV. — ELEMENTS   OF  THE    SATELLITES.    463 


Elements  of  the  Moon. 


Mean  distance  from  the  earth 69 . 96436  terrestrial  radii. 

Mean  sidereal  revolution „ 27.321661418  days. 

Mean  synodical  revolution 29.630588715  days. 

Mean  longitude  January  i,  1801 118°  17'    8". 3. 

Mean  longitude  of  perigee  at  do 266°  10'    7".  5. 

Mean  longitude  of  ascending  node  at  do 1 3°  53'  1 7".  7. 

Mean  inclination  of  orbit _ 5°    8'  47". 9. 

Mean  revolution  of  nodes 6798 .279  days. 

Mean  revolution  of  perigee 32,32 .676343  days. 

Eccentricity  of  orbit o.o548442. 

Diameter  of  the  moon 21 53  miles. 

Density,  that  of  the  earth  being  i o .  6667. 

Mass,  that  of  the  earth  being  i o.ou  399. 


Elements  of  the  Satellites  of  Jupiter. 


Sat. 

Sidereal  Revolution. 

Distance  in 
Radii  of  Jupiter. 

Orbit  inclined 
to  Jupiter's 
Equator. 

Diameter. 

Mass,  that  of  Jupi- 
ter being  1. 

Apparent. 

In  Miles. 

I 

2 

3 

4 

d.    h.     m.         s. 

i   18  27  33.5o5 
3   i3  i3  42.o4o 
7     3  42  33.36o 
16  16  32   11.271 

6.o4853 
9.62347 
i5.35o24 
26.99835 

o         /       i> 

007 
0      I      6 

o     5     3 

0        0    24 

I  .Ol5 
0.911 

1.488 
1.273 

2436 
2187 
3573 
3067 

.000017328 
.OO0023235 
.000088497 
.000042669 

Elements  of  the  Satellites  of  Saturn. 


Sat. 

Sidereal  Revolution. 

Distance  in 
Radii  of  Sat- 
urn. 

Eccentricity. 

Longitude  of 
Peri-Saturnium. 

Mean  Longitude. 

Epoch. 

d.    h.     Tfe.       s. 

0               /             » 

0               /           ft 

I 

o  22  36   17.7 

3.i4o8 

0.06889 

io4  42 

264  16  36 

1789.706 

2 

i     8  53     2.7 

4.o3i9 

Uncertain. 

67  56  26 

1789.706 

3 

i   21    18  33.o 

4.9926 

0.0061 

1  84  36 

i58  3i     o 

i836.3o8 

4 

2  17  44  5i  .2 

6.399 

O.02 

42  3o 

327  4o  48 

i836.o 

5 

4  12  26  ii  .  i 

8.932 

0.02269 

95 

353  44     o 

i836.o 

6 

i5  22  4i  24.9 

20.706 

0.029223 

244  35  5o 

137  21  24 

i83o.o 

7 

21       4    20 

26.029 

o.  n5 

296 

32 

1849.0 

8 

79    7  54  4o.8 

64.359 

269  37  48 

1790.0 

Elements  of  the  Satellites  of  Uranus. 


Sat. 

Sidereal  Revolution. 

Daily  Motion. 

Mean  apparent 
Distance. 

Mean  Distance  in 
Miles. 

I 

2 

3 

4 

Days. 

2.52o35 
4.14397 
8.706886 
13.463263 

0 

i42.8373 
86.8732 
4i.35i33 
26.73943 

i3.54 
19.28 
3i.44 
42.87 

119994 
170863* 
278627 
379921 

Elements  of  the  Satellite  of  Neptune. 

Sidereal  revolution 5d.  zih.  om.  1 75 

Apparent  mean  distance i6".75. 

True  mean  distance 232,ooo  miles. 

Orbit  inclined  to  the  plane  of  ecliptic 29°. 


464 


TABLE    XX XY. — ELEMENTS    OF   THE   ASTEROIDS. 


No 

Name. 

Discovered. 

£& 

3! 

Mean  Dis- 
tance. 

Eccentricity. 

Time  of  Si- 
dereal Rev- 
olution. 

When. 

By  whom. 

Where. 

Days. 

I 

Ceres 

1  80  1,  Jan.  i       jPiazzi 

Palermo 

8 

2.765765 

0.0791797 

l68o.o47 

2 

Pallas 

1802,  March  28  Gibers 

Bremen 

7 

2.769533 

0.239o447 

i683.48i 

3 

4 

Juno 
Vesta 

1804,  Sept.  i 
1807,  March  29 

Harding 

Lilienthal 
Bremen 

8 
6 

2.6686II 

2.36o559 

0.2565354 
0.0901807 

1592.  3o5 
1324.710 

5 

Astraea 

1  845,  Dec.  8 

Hencke 

Driessen 

9 

2  .577402 

0.1887517 

iSn.Syi 

6 

Hebe 

1847,  July  i 

Hencke 

Driessen 

9 

2.4245o3 

0.2022418 

1378.899 

7 

[ris 

1847,  Aug.  1  3 

Hind 

London 

8 

2.3865i4 

0.23l45l2 

1346.617 

8 

Flora 

1847,  Oct.  1  8 

Hind 

London 

8 

2.2oi386 

o.  1567040 

1193.007 

9 

Metis 

1  848,  April  26 

Graham 

Markree 

10 

2.386325 

0.1235524 

1346.457 

10 

Hygeia 

1849,  April  12 

Gasparis 

Naples 

9 

3.i49384 

o.ioo5585 

2o4l  .44l 

n 

12 

Parthenope 
Clio 

i85o,  May  ii 
i85o,  Sept.  1  3 

Gasparis 
Hind 

Naples 
London 

9 
9 

2.448097 
2.33468o 

0.0980302 
o.o454773 

1399.076 
1302.986 

i3 

i4 

Egeria 
'rene 

i85o,  Nov.  2 
i85i,  May  19 

Gasparis 
Hind 

Naples 
London 

9 
9 

2.576890 
2.584596 

0.0853696 
o.  1687952 

l5l2.  I  06 

1517.705 

i5 

Eunomia 

1  85  1,  July  29 

Gasparis 

Naples 

9 

2.6434io 

0.1878261 

1569.  8o3 

16 

3syche 

i852,  March  17 

Gasparis 

Naples 

10 

2.932951 

o.  1309376 

i834.66i 

jf| 

Thetis 

1  852,  April  17 

Luther 

Bilk 

IO 

2.4839o8 

o.i3o8o84 

1429.887 

1*8 

Melpomene 

1  852,  June  24 

Hind 

London 

9 

2.293580 

o  .2150762 

1268.729! 

J9 
20 

21 

Fortuna 
Vlassilia 
Lutetia 

I  852,  Aug.  22 

1  852,  Sept.  19 
1  852,  Nov.  1  5 

Hind 

Gasparis 
Goldschmidt 

London 
Naples 
Paris 

9 
9 
9 

2.444074 
2.401470 
2.434106 

o.  1587172 
o.i44665i 

0.  1624452 

1395.629 
1359.296 
1387.099 

22 

Calliope 

1862,  Nov.  16 

Hind 

London 

9 

2.911710 

o.io36i33 

1814.768 

23 

Thalia 

1862,  Dec.  i5 

Hind 

London 

10 

2.645i24 

0.2398050 

iSyi.SSi 

24 

Themis 

1  853,  April  5 

Gasparis 

Naples 

1  1 

3.144496 

O.  I22584l 

2036.690 

25 

Phocea 

i853,  April  6 

Chacornac 

Marseilles 

9 

2.400673 

o.253i266 

i358.6i9 

26 

Proserpina 

1  853,  May  5 

Luther 

Bilk 

IO 

2.587802 

0.0689696 

l522.  106 

27 

28 

Euterpe 
Bellona 

iSSSJNov.  8 
1  8  54,  March  i 

Hind 
Luther 

London 
Bilk 

9 

IO 

2.347853 
2.780725 

0.1714945 

0.  I62883I 

i3i4.Q29 
i693.693 

?9 
3o 

Amphitrite 
Urania 

1  8  54,  March  i 
1  8  54,  July  22 

Marth 
Hind 

London 
London 

10 

9 

2.546297 
2.358327 

o.o685i35 

0.1548981 

1484.096 
1322.834 

3i 

32 

Euphrosyne 
Pomona 

1  8  54,  Sept.  i 
1  854,  Oct.  28 

Ferguson 
Goldschmidt 

WWgton 
Paris 

9 

10 

3.  192287 
2.585o56 

0.2294184 
0.0956894 

2083.297 
i5i8.  109 

33 

Polymnia 

1  854.  Oct.  28 

Chacornac 

Paris 

9 

2.378790 

0.2243941 

1339.902 

tt 

TABLE    XXX  Y. — E  LEMENTS   OF   THE   ASTEROIDS.       465 


No 

Longitude  of 
Perihelion. 

Longitude  of 
ascending  Node 

Inclination  of 
Orbit. 

Mean  daily 
Motion. 

Mean  Longitude 
at  Epoch. 

Epoch,  mean  Berlin 
Time. 

I 

149  33  40.9 

80   48    22.1 

10    36    27.  L 

771.40703 

o         /            // 

97  39  57.- 

1  856,  July  i  .0 

2 

122      3   49.8 

172  38  20.  i 

34  42  37.4 

769.83370 

73  17     o.c 

1  856,  June  21.0 

3 

54     8  16.9 

170  5g  3i.8 

i3     3  27. 

8l3.9l433 

34  i    17  34.6 

i856,  Aug.  4.0 

4 

25o  46  10.^: 

io3  23  39.0 

7     8   17. 

978.32746 

83  55   i4.g 

i856,  Dec.  14.0 

5 

i35  4s  3i.7 

i4i  27  4y-5 

5   19  23. 

857.  49958 

197  37     6.S 

1  85  1,  April  29.  5 

6 

i5   10  35.^ 

i  38  33  19.4 

i4  46  4i. 

939.87995 

2i5  33     6.2 

1  854,  April  18.0 

7 

4i   :5  49-3 

259  45  58.3 

5  28  12. 

962.41141 

282  25  43.2 

1  854,  June  i3.o 

8 

32  54  28.3 

no   17  48.6 

5  53     8. 

1086.33098 

68  48  3i.9 

1  848,  Jan.  i.o 

9 

71   38  35.2 

68  3o  28.0 

5  35  34. 

962.52560 

97  4o  37.7 

1  854,  July  i  .0 

10 

227     3     5.7 

287  38  24.4 

3  47  10. 

634.84564 

354       2    42.9 

i85i,  Sept.  17.0 

ii 

3i7     3  5o.6 

124  59  53.6 

4  36  54. 

926.32568 

86     2  56.o 

i85a,  July  i3.o 

12 

3oi   52  59.0 

235  29  28.2 

8  23     6. 

994.63869 

7  4i     3.9 

1  85  1,  Jan.  o.o 

i3 

119  36  46.4 

43  18  46.6 

16  32  Sg. 

857.08269 

229  43  .  5.4 

i852,  Dec.  21  .0 

i4 

178  46  5o.o 

186  48  56.3 

9     6  44.4 

853.92120 

224    22    27.0 

i85i,  May  21.0 

i5 

27  53  58.3 

293  54  3o.o 

ii  43  59-6 

825.58i34 

342  i5  53.8 

i852,  Jan.  o.o 

16 

ii   28     9.2 

i5o  36  43.6 

3     3  36.  7 

706.39770 

149  20  53.^ 

1  852,  March  3i.o 

*7 

258  33  4i-7 

125   19  33.2 

5  35  45.9 

906.36543 

333  45  47.7 

1  853,  Sept.  17.0 

18 

i5  3o  17.3 

149  58  49.8 

10     9  37.5 

IO2I  .495OO 

3oi   25  52.3 

1  852,  July  7.  5 

19 

3o  48     6.8 

2i!   26  56.5 

i   32  27.4 

928.6l35o 

6  12  43.7 

1  852,  Nov.  5.o 

20 

99     2  17.4 

206  56  49.2 

O    4l     II  .2 

953.43496 

44  37  39.4 

1  853,  Jan.  o.o 

21 

326  33     2.1 

80  26  5o.2 

3     5  36.  9 

934.324i3 

43  32  5i.5 

1  853,  Jan.  9.0 

22 

58  49  24.2 

66  36  5o.5 

i3  44  48.7 

714.14280 

77     6  24.4 

1  853,  Jan.  o.o 

23 

122  44  39.2 

67  53  29.0 

10  i3  54.8 

824.77865 

89  i5     5.4 

1  853,  Jan.  o.o 

24 

i  34  43  45.i 

35  4o  35.5 

o  49  29.6 

636.3265o 

214  34  21.4 

1  854,  Jan.  1.0372 

25 

3o2  37  21.6 

214     3  4o.o 

21   36     5.3 

953.91000 

265    22    24.8 

1  853,  July  i.o 

26 

175     9  42.9 

45  36  47.3 

3  43  20.1 

85i  .452oo 

216  Sg  42.6 

1  853,  June  o.o 

27 

87  i5  29.0 

93  3o  18.7 

i   36     9.0 

986.28010 

74  46  25.6 

1  854,  Jan.  1.0372 

28 

119  38  4i  •  i 

44  5  i     9.7 

9  25     6.8 

765  .  19200 

i57  5i  46.2 

1  8  54,  March  o.o 

29 

54    4  26.2 

356   i5  54.6 

6     4     6.3 

873.25800 

181   28  23.i 

1  8  54,  March  o.o 

3o 

26  42  59.3 

'07  57  5i.i 

i  56  4i  '7 

979.71500 

324  56  16.7 

i854,  July  22.03-72 

3i 

95  i3  45.i 

3i    ii   59.9 

26  53  a6.o 

622  .09100 

34  i  3  48.5 

1  854,  Sept.  i.o 

32 

195  46  56.o 

220    44    2O.  5 

5  39     2.9 

853.69400 

42    22    23.6 

i854,  Nov.  o.o 

33 

22  25  58.4 

I     12    29.2 

I     22    2O.  6 

967.23500 

32    52       6.9 

1  8  54,  Nov.  o.o 

466 


TABLE    XXXVI. 


For  Sines  and  Tangents  of  small  Arcs. 


Arc. 

Log.  Sine. 

Log.  sin.  A  —  log.  A".j 

Diff.llT.I   Log.  Tangent. 

Log.  tan.  A—  log.  A".|Diff.  10". 

Arc. 

o        / 

0            / 

O       O 

Inf.  Neg. 

4.6855748,7 

Inf.  Neg. 

4.6866748,7 

0       O 

I 

6.4637261 

5748,6 

O,OI 

6.4637261 

5748,8 

0,02 

'nfi 

I 

2 

6.7647661 

5?48,4 

,o3 

6.7647662 

67-49,2 

.Ou 

2 

3 

6.94o8473 

5748,i 

} 

6.94o8475 

5749^8 

,10 

3 

4 

7.0667860 

6747,7 

;°7 

7.0667863 

575o.6 

, 

18 

4 

o     5 

7.  1626960 

4.6866747,1 

?°9 

7.  i626964 

4.685575i!7 

,10 
O  22 

o     5 

6 

7.2418771 

6746,6 

0,11 

T    Q 

7.2418778 

5753'.i 

6 

7 

7.3o88239 

5745,7 

7 

7.3088248 

5754^ 

'o^ 

7 

8 

7.3668167 

5744,7 

,16 

7.3668169 

5756,5 

.01 

35 

8 

9 

7.4i7968i 

5743,7 

3*7 

7.4179696 

5758,6 

9 

o   10 

7.4637255 

4.6866742,6 

O  2  I 

7.4637273 

4.6866760,9 

'/3 

0     10 

1  1 

7.5o5u8i 

574i,3 

7.5o5i2o3 

5763,5 

/ 

ii 

12 

7.5429o65 

5739,8 

'ofi 

7.6429091 

6766.3 

^ 

12 

i3 

7.6776684 

«p      5738,3 

'28 

7.6776715 

6769^4 

'55 

i3 

i4 

7.6o9853o 

>r 

Xn 

7.6098666 

6772,7 

', 

i,4 

o  i5 

7.6398i6o 

4.6855734'9 

,00 

7.6398201 

4.6866776,2 

o'.63 

o   16 

16 

7.6678445 

6733,0 

0,02 

Q  / 

7.6678492 

6780^0 

fin 

16 

I7 

7.694i733 

573i,o 

04 
Qfi 

7.6941786 

6784'! 

?D7 

17 

18 

7.7189966 

5728^ 

,00 

QQ 

7.  7190026 

6788^4 

'7g 

18 

19 

7.7424776 

6726  6 

,00 
/in 

7.7424841 

6792.9 

'g0 

19 

0    20         7.7647637 

4.6866724,2 

,40 

7.7647610 

4.6866797.7 

Q  / 

0    20 

21 

r.7859427 

572i,7 

O,42 

7.7869608 

6802^ 

88 

21 

22 

7.8o6i458 

6719  o 

/.p. 

7.8061647 

58o8'o 

,00 

22 

23 

7.  8264607 

57i6,3 

,40 

7.8264604 

58i3;5 

nfi 

23 

24 

7.8439338 

,48 

7.8439444 

68192 

,90 

24 

o  26 

7.8616623 

4.68557io'4 

'   ° 

7.8616738 

4.6866826  2 

I  ,OO 

0    25 

26 

7.8786963 

5707,3 

0,02 

7.8787077 

563*5 

1,04 

26 

27 

7.8960864 

57o4,o 

5« 

7.895o988 

5838o 

3 

27 

28 

7.9108793 

57oo  6 

>?I 

7.9io8938 

5844.7 

tfi 

28 

29 

7.9261  190 

5697-8 

'fin 

7>926i344 

585i'7 

,10 

29 

o  3o 

7.9408419 

4.6855693  5 

,00 

O62 

7>94o8584 

4.68558589 

.20 

o  3o 

3i 

7.9660819 

6689,8 

7.955o996 

5866,4 

J:2 

3i 

32 

7.9688698 

5686,o 

fifi 

7.9688886 

5874,i 

,29 

QQ 

32 

33 

7.9822334 

6682,0 

,OO 

7-9822534 

6882,1 

,00 

33 

34 

7.9961980 

66779 

I 

7-9952I92 

589o3 

/T 

34 

o  35 

8.0077867 

4.68556736 

?7° 

8.oo78o92 

4.6855898,7 

,4  1 

o  35 

36 

8.0200207 

5669'3 

fc 

8.0200445 

59o74 

'/o 

36 

37 

8.0319196 

5664,8 

;7D 

8.03i9446 

6916,4 

,49 

Co 

37 

38 

8.0436009 

566o  2 

;77 

8.o435274 

6926^6 

.00 

38 

39 

8.0647814 

5655.5 

;79 

QT 

8.o548o94 

6935  o 

fir 

39 

o  4o 

8,o657763 

4.685565o'7 

,01 

8.0668067 

4.6855944,7 

,01 

o  4o 

4i 

8.0764997 

5645.7 

o,83 
ok 

8.0766306 

5954,6 

'fin 

4i 

42 

8.0869646 

564o'6 

,00 

8.o86997O 

5964,8 

,09 

nA 

42 

43 

8.0971832 

5635,4 

>  7 

8.o972i72 

5975,2 

?74 

43 

44 

8.  1071669 

563o,i 

'  9 

8.  1072026 

5985,8 

82 

44 

o  45 

8.  1169262 

4.6866624,6 

Or\  ^ 

8.n69634 

4.6855996,7 

T  86 

o  45 

46 

8.1264710 

6619.1 

,(JO 

nc 

8.  i265o99 

6007,9 

1  .OU 

46 

47 

8.i358io4 

56i3!4 

,9D 

8.i3585io 

6019,3 

,90 

n  ft 

47 

48 

8.1449532 

6607,6 

397 

8.1449966 

6o3o.9 

,94 

48 

49 

8.1639076 

6601,6 

;99 

8.i5395i6 

6042^8 

;9 

49 

o  5o 

8.1626808 

4.6855595,5 

1,01 

8.  i627267 

4.6866064^9 

2,02 

o  5o 

5i 

8.  1712804 

5589,4 

''oS 

8.i7i3282 

6067,3 

T  C\ 

5i 

62 

8.  1797129 

5583,i 

,00 

8.1-79-7626 

6079,9 

.10 

T  /! 

62 

53 

8.1879848 

6676,6 

>°7 

8.i88o364 

6092,8 

.  14 

53 

54 

8.  1961020 

6670,1 

3°9 

8.1961666 

6106,  9 

'l? 

54 

,o  55 

8.2040703 

4.6855563,4 

?I^ 

8  .2041269 

4.6866119,2 

,20 

o    o  rr 

o  55 

56 

8.2118949 

5556,6 

i,U 

8.2119626 

6i32,8 

^3^  / 
QT 

56 

67 

8.2196811 

5549,7 

,i5 

8  2196408 

6i46,7 

'35 

57 

58 

8.2271335 

5542,6 

;r7 

8.2271963 

6160,8 

,00 

58 

59 

8.2345568 

5535,5 

;X9 

8.2346208 

6i75',i 

XT 

59 

o  60 

8.24i8553 

4.6855528,2 

.2  I 

8.2419216 

4.6866189,7 

,40 

o  60 

TABLE    XXXVI. 


For  Sines  and  Tangents  of  small  Arcs. 


467 


Arc. 

Log.  Sine. 

Log.  sin.  A  —  log.  A" 

Diffi  10" 

Log.  Tangent. 

Log.  tan.  A  —  tan.  A" 

UifflO".!     Arc. 

0             / 

I        0 

8.24i8553 

4.6855528,2 

I    0/1 

8  .  24i92i5 

4.6856i89,7 

I        0 

I 

8.249o332 

5520,8 

'26 

8.249ioi5 

6204,5 

2,47 

^  T 

I 

2 

8.256o943 

55i3,2 

'28 

8.256i649 

62i9,6 

55 

2 

3 

8.263o424 

55o5,6 

,20 

8.263n53 

6234,9 

5Q 

3 

1       4 

8.26988io 

5497,8 

,3o 

8.2699563 

7^ 

63 

4 

i     5 

8.2766i36 

4.6855489,9 

i  1A 

8.27669i2 

4.6856266,2 

.UO 

2  68 

i     5 

6 

8.2832434 

548i,9 

i  .  O<4 

8.2833234 

6282,3 

6 

7 

8.2897734 

5473,7 

'38 

8.2898559 

6298  6 

'76 

7 

8 

8.2962o67 

5465,5 

Ao 

8  .29629i7 

63i5  i 

'fin 

8 

9 

8.3o2546o 

5457,i 

,4o 

/o 

8.3026335 

633i.9 

»0 

Q/. 

9 

I     10 

8.3o8794i 

4.6855448,6 

,42 

8.3o8884a 

4.6856348,9 

,04 

I     10 

ii 

8.3i49536 

5439,9 

/fi 

8.3i5o462 

6366,2 

J 
CtO 

ii 

12 

8.32io269 

543i.2 

40 

/  O 

8.32II22I 

6383  7 

j92 

12 

i3 

8.3270163 

5422^3 

,48 

8.327II43 

64oij5 

* 

i3 

i4 

8.3329243 

54i3,3 

K0 

8.333o249 

64i9  5 

0,00 
o/i 

i4 

i   i5 

8.3387529 

4.68554o4,i 

* 
T'c  / 

8.3388563 

4.6856437:8 

,04 

3  08 

i  i5 

16 

8.3445o43 

5394,9 

1  ,04 

8.3446io5 

6456,3 

0,00 

16 

17 

8.35oi8o5 

5385,5 

'co 

8.35o2895 

6475,0 

?12 

17 

18 

8.3557835 

5376,o 

,58 

8.3558953 

6494,o 

1ll 

18 

i9 

8.36i3i5o 

5366,4 

3fi 

8.36i4297 

65i3,2 

.21 

i9 

I     20 

8.3667760. 

4.6855356,7 

fif 

8.3668945 

4.6856532,7 

,25 
"3.   r>n 

I    20 

21 

8.3721710 

5346,8 

I  »t)A 
66 

8.37229i5 

6552,5 

0,29 

21 

22 

8.3774988 

5336,8 

j°° 

8.3776223 

6672,4 

' 

22    ! 

23 

8.3827620 

5326,7 

5 

8.3828886 

6592.6 

/  I 

?3 

24 

8.3879622 

53i6,5 

'7o 

8.388o9i8 

66  1  3.  i 

,41 

24 

I     25 

8.393ioo8 

4.68553o6,i 

a 

8.3932336 

4.6856633,8 

3  Act 

I     25 

26 

8.3981793 

5295,7 

1,70 

8.3983i52 

6654,8 

o,49 

26 

27 

8.4031990 

5285,! 

?77 

8.4o3338i 

6676,0 

'57 

27 

28 

8.4o8i6i4 

5274,4 

'8? 

8.4o83o37 

6607,4 

61 

28 

29 

8.4130676 

5263,5 

'83 

8.4i32i32 

8719,1 

,01 
66 

29 

i    3o 

8.4179190 

4.6855252,6 

,00 

8.4i8o679 

4.6856741,0 

,00 

3  70 

i   3o 

3i 

8.4227168 

524  1,  5 

r> 

8.422869o 

6763,2 

3i 

32 

8.4274621 

523o,3 

fin 

8.4276176 

6785,6 

'  8 

32 

33 

8.432i56i 

52i8,9 

J°9 

8.4323i5o 

68o8,3 

Qn 

33 

34 

8.4367999 

5207,5 

'o3 

8.4369622 

683i,2 

02 
fifi 

34 

i   35 

8.44i3944 

4.6855i95,9 

,9J 

T     r\  S 

8.44i56o3 

4.6856854,4 

,00 

i   35 

36 

8.44594o9 

5i84J2 

I,9D 

8.446no3 

6877,8 

'0/ 

36 

37 

8.45o44o2 

5172,4 

;97 

8.45o6i3i 

69oi,4 

:94 

37 

38 

8.4548934 

5i6o.4 

;99 

2r\  T 

8.455o699 

6925,3 

f 

38 

39 

8.4593oi3 

5i48,4 

.0  1 

8.45948i4 

6949.5 

4,O2 

39      ; 

i  4o 

8.4636649 

4.6855i36,2 

,o3 

8.4638486 

4.6856973,9 

>° 

4i 

8.467985o 

5i23,9 

2  OD 

8.468i725 

6998,5 

'i" 

4i 

42 

8.4722626 

5in,4 

5°7 

8.4724538 

7023,4 

? 

42 

43 

8.4764984 

5o98,9 

j°9 

8.4766933 

7o48,5 

jl? 

43 

44 

8.48o6932 

5o86,2 

5     * 

8.48o892o 

7073,8 

0  "7 

44 

i  45 

8.4848479 

4.6855o73,4 

J 

8.485o5o5 

4.6857o99,5 

527 

4,3i 

i  45 

46 

8.4889632 

5o6o,5 

>* 

8.489i696 

7i25,3 

35 

46 

47 

8.493o398 

5o47,4 

?*7 

8.49325o2 

7i54,4 

,OJ 

3o 

47 

48 

8.4970784 

5o34,3 

?T9 

8.49.72928 

7177,8 

43 

48 

49 

8.5oio798 

5O2I,O 

,22 

8.5oi2982 

7204,4 

,40 

49 

i  5o 

8.5o5o447 

4.6855oo7,6 

fi 

8.5052671 

4.6857231,2 

/?5t 

i   5o 

5i 

8.5o89736 

4994,o 

2.2O 

* 

8.5o92ooi 

7258,3 

,55 

5i 

52 

8.5i28673 

4980,4 

'q 

8.5i3o978 

7285,6 

52 

53 

8.  5i67264 

4966,6 

,00 

8.5i696io 

73i3,2 

?64 

53 

54 

8.52o55i4 

4952,7 

'*A 

8  .  52O79O2 

734i,o 

,V7£f 

an 

54 

i  55 

8.  524343o 

4.6854938.6 

,34 

8.524586o 

4.6857369,i 

,00 

/    r*2 

i  55 

56 

8.5281017 

4924,5 

2.36 

8.528349o 

7397,4 

'76 

56 

57 

8.53i828i 

4910,2 

5 

8.5320797 

7425,9 

?/ 

80 

57 

58 

8.5355228 

4895,8 

,40 

8.5357787 

7454,7 

,00 

58 

59 
i  60 

8.539i863 
8.5428192 

488i,3 
4.6854866,7 

M 

8.5394466 
8.543o838 

7483,8 
4.68575i3,i 

IBS 

59 
i  60 

468  TABLE   XXXVII. — NUMBERS  OFTEN   USED  IN   CALCULATIONS. 


Constants. 

Logarithms. 

Area  of  a  circle  to  radius  i                      } 

Circumference  of  a  circle  to  diameter  i  >=     T      =           3.  i4i  59,26536 

0.4971499 

Surface  of  a  sphere  to  diameter  i            3 

Area  of  a  circle  to  diameter  i                     =     TT—  4=           0.78539,81684 

9.8960899 

Capacity  of  a  sphere  to  diameter  i              =     v—6=           o.  62869,  8  7756 

9.7189986 

Capacity  of  a  sphere  to  radius  i                 =  4^—3—          4.18879,02048 

0.6220886 

Diameter  of  a  circle  to  area  i                     —-\/k—n—           1.12887,91671 

O.o52455l 

)iameter  of  a  sphere  to  capacity  i             r=-\/6—  TT—           i  .24070,09818 

0.0936671 

VTT       =             1.77245,38609 

0.2486749 

7T3        =                 9.86960,44011 

0.9942997 

I-7-7T=                 0.3l83o,98862 

9.6028601 

i-=-7r2—           o.  10182,  u836 

9.0067008 

3ase  of  Naperian  logarithms                      —    e        =           2.71828,18286 

0.4342945 

Modulus  of  the  common  logarithms            =    M       =           0.48429,44819 

9.6377843 

Naperian  logarithm  of  TT                                               —           1.  14472,  9  8858 

0.0687030 

£  degrees                =         57.29577,96180 

i  .7681226 

Arc  equal  to  radius  expressed  in  <  minutes                =     3487.74677,07849 

8.5362789 

'seconds                =206264.  80624,70964 

5.3i4425i 

jength  of  one  degree  in  parts  of  radius                     —             .01746,32926 

8.2418778 

jength  of  one  minute  in  parts  of  radius                                     .00029,08882 

6.4687261 

Sine  of  i  second                                                          =             .00000,48481 

4.6866749 

Sine  of  2  seconds                                                           =              .00000,96963 

4.9866049 

Sine  of  3  seconds                                                          =              .00001,  45444 

5.  1626961 

36o  degrees  expressed  in  seconds  of  arc                                            1296000 

6.  i  126060 

24  hours  expressed  in  seconds  of  time                        =                          864oo 

4.9866137 

12  hours  expressed  in  seconds  of  time                        =                          48200 

4.6354837 

lumber  of  feet  in  one  mile                                          —                            6280 

8.7226389 

Sidereal  year  in  mean  solar  days                               =       366.25687,4417 

2.6626978 

Tropical  year,  1860,  in  mean  solar  days                    =       866.24221,6787 

2.6626810 

Annual  variation                                                            =         —  .00000,00669 

Sidereal  rotation  of  earth  in  mean  solar  seconds        =  86164.09966,888 

4.9868264 

Sidereal  (*'.  e.9  equin.)  day  in  mean  solar  seconds     =  86164.09064,9806 

4.9353263 

VIean  solar  day  in  seconds  of  sidereal  time                =  86636.  55534,883i4 

4.9877012 

Acceleration  of  stars  in  solar  day=  |  ^^gj  ^t^tta^' 

2.8739828 
2.8727453 

Compression  of  the  earth—  1-4-299.  162818 

7.6241069 

Equatorial  radius  of  the  earth  in  English  feet                            20923699.98 

7.32o636^ 

3olar  radius  of  the  earth  in  English  feet                   =              20863667.16 

7.8191823 

degree  of  latitude  at  the  equator  in  English  feet      —                  862748.83 

5.5596o54 

Degree  of  latitude  at  45°  in  English  feet                   =                  864671  .77 

5.56i783o 

£      French  toises  into  French  metres                      =           i  .  94908669 

s^     fo.  2898200 

French  toises  into  English  yards                       =           2.1  8168084 

J       0.8286916 

<2  M  French  toises  into  English  feet                           —          6.39469262 

"3  «    0.8068129 

5  -    French  feet  into  English  feet                             =           i  .06676642 

|S    o.o2767i6 

3,>    French  metres  into  English  yards                       =           1.098683067 

|1    o.o3887i6 

1      French  metres  into  English  feet                         =           8.280899167 

S,      0.6169929 

S     [French  metres  into  English  inches                     =         89.87079 

j     [i.595i74i 

British  imperial  gallon  in  cubic  inches                       =:       277.274 
Cubic  inch  distilled  water  in  grains  (B.  3o  in.  T.  62°)—       262  .458 

2.4429090 
2.4021891 

{  in  Ibs.  avoirdupois    =         62.32106067 

i.  7946348 

in  oz.  avoirdupois     =       997.18696914 

2.998-7648 

in  Ibs.  Troy             =         76.7874 

i  .8793io4 

in  oz.  Troy               —       908  .  8488 

2.9684916 

ingrains                   —486247.424 

5.6397329 

Length  of  seconds  pendulum  in  inch-     p°n-                       o^'1  ^2^ 

i  .6926129 

GS  nt                                                                 x^<iris           —  .          o^«  12^20 

i  .  6926019 

NewYork=;        89.1012 

1.6921901 

Velocity  of  falling  bodies  in  inches  in     £°ndon      =       ***  '  289^ 
one  second  at                                           Pans                    386.  19061 

2.6869126 
2.6868016 

New  York—       385.  9i  337 

2.6864898 

EXPLANATION  OF  THE  TABLES. 


TABLE  I.,  page  357,  contains  the  Latitudes  and  Longitudes  of 
the  principal  foreign  Observatories,  taken  chiefly  from  the  Ameri- 
can Nautical  Almanac.  In  several  cases,  these  numbers  differ 
slightly  from  those  given  in  the  English  Nautical  Almanac  and 
the  Berlin  Jahrbuch. 

TABLE  II.,  page  358,  contains  the  Latitudes  and  Longitudes 
of  various  places  in  the  United  States.  This  list  is  designed  to 
embrace  the  large  cities,  the  astronomical  observatories,  and  the 
principal  colleges  of  the  country.  A  few  of  the  determinations 
are  derived  from  the  observations  of  the  United  States  Coast 
Survey  ;  others  have  been  derived  from  the  labors  of  numerous 
private  observers ;  while  many  have  been  taken  from  maps  which 
are  confessedly  very  imperfect.  It  is  hoped  that  before  many 
years  this  Table  may  be  very  much  improved. 

TABLE  III.,  page  359,  serves  to  convert  hours,  minutes,  and 
seconds  into  decimals  of  a  day,  and  vice  versa. 

Example  1.  It  is  required  to  convert  14h.  17m.  16.4s.  into 
the  decimal  of  a  day. 
We  find  from  the  Table, 

14h.  =.5833333 
17m.  =  .0118056 
16s.  =.0001852 
0.4s.  =.0000046 

Hence          14h.  17m.  16.4s.  =  .5953287 
The  equivalent  for  0.4s.  is  derived  from  the  equivalent  for  4s. 
by  removing  the  decimal  point  one  place  to  the  left. 

Example  2.  Let  it  be  required  to  convert  0.5953287  day 
into  hours,  minutes,  and  seconds.  We  find  from  the  Table, 


470  PRACTICAL    ASTRONOMY. 


.59           =14h. 

9m.  36s. 

.005 

7      12.0 

.0003       = 

25.92 

.00002     = 

1.73 

.000008  = 

.69 

.0000007  = 

.06 

Hence  .5953287  =  14h.  17m.  16.40s. 

The  number  of  seconds  corresponding  to  .00002  is  obtained 
from  the  little  table  of  proportional  parts  at  the  bottom  of  page 
359.  Thus,  if  .0002  is  equivalent  to  17.28s.,  .00002  must  be 
equivalent  to  1.728s. ;  and  we  may  proceed  in  the  same  manner 
for  other  fractions. 

TABLE  IV.,  page  360,  serves  to  convert  intervals  of  mean  solar 
time  into  equivalent  intervals  of  sidereal  time. 

Example.  It  is  required  to  find  the  sidereal  interval  corre- 
sponding to  the  mean  solar  interval,  2h.  22m.  25.62s. 
2h.   Om.    Os.      solar  interval  equals  2h.    Om.  19.713s.  sid.interv. 

22       0  «  "  22       3.614          " 

25  «  "  25.068          " 

0.62          "  "  0.622          " 

2h. 22m.  25.62s.  solar  interval  equals  2h.22m.497til7sL  sid.  interv. 

The  method  of  converting  mean  solar  time  into  sidereal  time 
is  explained  on  page  123. 

TABLE  V.,  page  361,  serves  to  convert  intervals  of  sidereal 
time  into  equivalent  intervals  of  mean  solar  time. 

Example.  Find  the  mean  solar  interval  corresponding  to  the 
sidereal  interval,  2h.  22m.  49.02s. 

2h.   Om.   Os.       sid.  interval  equals  Ih.  59m.  40. 341s.  solar  interv. 
22       0  «  «  21     56.396 

49  "  "  48.866  « 

0.02  "  "  0.020  " 

2h.  22m.  49.02s.  sid.  interval  equals  2h.  22m.  25.623s.  solar  interv. 
The  method  of  converting  sidereal  time  into  mean  solar  time 
is  explained  on  page  125. 

TABLE  VI.,  page  362,  serves  to  convert  degrees,  minutes,  and 
seconds  of  space  into  hours,  minutes,  and  seconds  of  time.  It  is 


EXPLANATION   OF   THE    TABLES.  471 

founded  on  the  ratio  of  15  degrees  to  1  hour.  The  Right  As- 
censions of  the  heavenly  bodies  are  sometimes  expressed  in  arc, 
but  generally  in  time. 

Example.  The  Right  Ascension  of  a  Lyrse  for  January  1, 
1855,  is  277°  59'  51".60.  Required  its  Right  Ascension  ex- 
pressed in  time. 

The  Equivalent  in  time  for  277°    0'  0"    is  18h.  28m.    Os. 
"  "  59  0       "  3      56 

«  "  51       "  3.40 

"  "  0.60  "  0.04 

The  Right  Ascension  in  time  is  18h.  31m.  59.44s. 

In  taking  out  the  equivalents  for  tenths  of  seconds  of  space, 
we  may  use  the  units  in  the  seconds  column  as  arguments, 
taking  care  to  remove  the  decimal  point  of  the  corresponding 
equivalent  one  place  to  the  left.  Thus  the  equivalent  for  6"  is 
0.4s.,  and  for  0".6  the  equivalent  is  0.04s. 

TABLE  VII.,  page  363,  serves  to  convert  hours,  minutes,  and 
seconds  of  sidereal  time  into  degrees,  minutes,  and  seconds  of 
space. 

Example.  The  Right  Ascension  of  a  Lyrae  for  January  1, 
1855,  is  18h.  31m.  59.44s.  Required  its  Right  Ascension  ex- 
pressed in  arc. 

The  Equivalent  in  arc  for  18h.    Om.  Os.     is  270°    W    0" 
«  "  31      0       "       7    45     0 

«  «  59  14  45 

«  «  0.44  "  6  .60 

The  Right  Ascension  in  arc  is  277°  59/  51/x.60 

TABLE  VIII.,  pages  364-5,  furnishes  the  amount  of  atmos- 
pheric refraction  for  all  altitudes  from  the  horizon  to  the  zenith. 

This  Table  was  constructed  by  the  late  Professor  Bessel,  of 
Konigsberg,  and  is  now  more  generally  used  than  any  other. 

It  requires,  in  addition  to  the  observed  apparent  altitude,  an 
observation  of  the  height  of  the  barometer,  upon  which  depends 
the  factor  B  ;  of  the  thermometer  attached  to  the  barometer, 
upon  which  depends  the  factor  t ;  and  of  the  temperature  of  the 
external  air,  upon  which  depends  the  factor  T.  If  the  attached 


472  PRACTICAL   ASTRONOMY. 

thermometer  is  not  observed,  we  may  assume  that  its  indica- 
tions are  the  same  as  those  of  the  external  thermometer. 

The  refraction  may  be  computed  either  by  natural  numbers 
or  by  logarithms.  The  latter  method  is  the  most  accurate,  as 
the  corrections  required  for  small  altitudes,  indicated  by  the  fac- 
tors M  and  N,  can  be  conveniently  applied  only  with  logarithms. 
When  the  altitude  is  not  very  small,  and  the  greatest  accuracy 
is  not  required,  the  use  of  logarithms  may  be  dispensed  with. 

By  natural  Numbers. 

From  the  accompanying  Table,  take  the  mean  refraction  cor- 
responding to  the  observed  altitude ;  take  the  factor  B,  corre- 
sponding to  the  height  of  the  barometer ;  also,  take  the  factor  £, 
corresponding  to  the  attached  thermometer,  and  the  factor  T, 
corresponding  to  the  external  thermometer.  Multiply  these  four 
numbers  together,  and  you  will  obtain  the  true  refraction. 

Example.  The  observed  apparent  altitude  of  a  star  was  34° 
ll/  15X/ ;  the  barometer,  28.856  inches ;  the  external  and  the 
attached  thermometers  both  stood  at  + 19.6°  Fahr.  It  is  re- 
quired to  compute  the  refraction. 

Mean  refraction  for  34°  II'  15'" V  24".8. 

Barometer,  28.856  ;      Factor  B,  0.975. 
Factor  £    1.001. 


Thermometer,  19.6°  . 

(  Factor  T,  1.061. 

Product,         0.975  x  1.001  x  1.061   =  1.0355. 
True  refraction  =  84".8x  1.0355  =  V  27x/.8. 

By  Logarithms. 

Take  from  the  Table  the  factor  log.  B,  corresponding  to  the 
height  of  the  barometer ;  also  the  factors  log.  t  and  log.  T,  cor- 
responding to  the  attached  and  external  thermometers.  Take 
also  the  values  of  log.  A,  as  also  M  and  N,  corresponding  to  the 
apparent  altitude.  Multiply  the  sum  of  log.  B  and  log.  t  by  M  ; 
also,  multiply  log.  T  by  N.  Take  the  algebraic  sum  of  these 
products  (regard  being  had  to  their  signs),  and  add  to  it  log.  A, 
and  the  logarithmic  cotangent  of  the  apparent  altitude.  The 
sum  will  be  the  logarithm  of  the  refraction  expressed  in  seconds 
of  arc.  This  rule  is  expressed  more  concisely  thus : 


EXPLANATION   OP  THE    TABLES. 


473 


The  logarithm  of  the  refraction  is 

=log.  cotangent  app.  alt.  +  log.  A+M(log.  B+log.  tf)  +  N  log.  T. 
Example  I.  The  observed  apparent  altitude  of  a  star  was  3° 
44X  40" ;  the  barometer,  30.162  inches  ;  the  attached  thermom- 
eter, 52.2°  Fahr. ;  and  the  external  thermometer,  46.6°  Fahr. 
Required  the  refraction. 

log.  factor  B,  30.162 
log.  factor  t,    52.2°  Fahr. 
log.  factor  T,  46.6°  Fahr. 


Apparent  altitude,  3°  44'  40" 
3°  44'  40" 


+  0.00821 
-0.00078 
+  0.00183 
M  =  1.0187 
N  =  1.1753 


log.  cotang.,  3°  44'  40"  1.18412 

log.  A,  1.68084 

1.0187 x (log.  B+log.  0,  +0.00757 

1.1753  x  log.  T,  +0.00213 

log.  refraction,     2.87466 
Refraction         =  127  29".3. 

Example  2.  The  observed  apparent  altitude  of  a  star  was  6° 
46'  40" ;  height  of  the  barometer,  29.772  inches  ;  the  attached 
thermometer,  —  0.4°  Fahr. ;  and  the  external  thermometer, 
—  2.0°  Fahr.  Required  the  refraction. 


log.  .cot.  6°  46'  40" 
log.  A 
log.  B             =+0.00256 
log.*              =+0.00127 
log.  B  +  log.  t  =+0.00383 
M(log.  B  +  log.  t) 
log.  T             =+0.04545 
N  log.  T 
log.  refraction, 
The  re  fraction  = 

0.92500 
1.73061 

0.00386 
0.04906 

M  =  1.0079 
N  =  1.0794 

2.70853 
8'  31".13 

TABLE  IX.,  page  366-7,  contains  the  coefficients  for  computing 
the  corrections  required  for  transit  observations  at  the  latitude 
of  Washington  Observatory. 

The  column  headed  Azimuth  contains  the  value  of  the  factor 
sin.  (0— 6)  sec.  6,  computed  for  </>  =  38°  53'  39"  for  all  altitudes 
from  the  south  horizon,  corresponding  to  a  north  polar  distance 
140°,  to  the  north  horizon,  corresponding  to  north  polar  distance 


474  PRACTICAL   ASTRONOMY. 

—  38°.  The  column  headed  Level  contains  the  value  of  the 
factor  cos.  (0— 6)  sec.  (5,  in  the  same  manner  for  every  degree 
of  altitude ;  and  the  column  headed  Collimation  contains  the 
value  of  sec.  d.  Near  the  pole,  the  values  of  these  coefficients 
change  very  rapidly,  and  it  is  more  convenient  to  compute  spe- 
cial tables  for  such  stars  as  are  frequently  observed.  Page  367 
exhibits  the  form  of  such  tables  for  Polaris,  /I  and  6  Ursse  Mino- 
ris,  and  51  Cephei,  both  for  the  upper  and  lower  culminations. 

The  use  of  this  Table  has  been  sufficiently  explained  on  page 
73,  and  several  preceding  pages. 

TABLE  X.,  pages  368-371,  furnishes  the  reduction  to  the  me- 
ridian for  a  star  observed  a  few  minutes  before  or  after  its  me- 
ridian passage.  It  enables  us  to  compute  more  readily  the  cor- 
rection to  be  applied  to  the  zenith  distance  observed  near  the 
meridian,  in  order  to  obtain  the  true  meridional  zenith  distance. 
This  correction  may  be  put  under  the  following  form : 

.      cos.  d>  cos.  6     -r,     /cos.  (b  cos.  6\2 

x= A  x  —  B  x  I £ )  x  cot.  z, 

sin.  z  \      sin.  z     ) 

,  ,     2  sin.2  JP        ,  B     2  sin.4  JP 

where  A= — : — — -,  and  B  —  — ; — T~-. 

sin.  V  sm.  V 

2  sin  2  AP 

Part  I.  shows  the  value  of  the  factor  -  '  ^  ,  and  the  argu- 
ment of  the  table  is  the  distance  in  time  of  the  sun  or  star  from 
the  meridian.  This  value  (or  the  sum  of  those  values  divided 
by  the  number  of  observations,  if  more  than  one  observation  has 

been  made)  must  be  multiplied  by  -  — ,  and  the  product 

subtracted  from  the  zenith  distance  (corrected  for  refraction,  etc.) 
of  the  sun  or  star  observed  near  the  meridian.  The  difference 
thus  obtained  will  give  the  true  meridional  zenith  distance  of 
the  sun  or  star  as  correctly  as  if  it  had  been  observed  precisely 
on  the  meridian.  dim. 

When,  however,  the  distance  from  the  meridian  is  considera- 
ble, and  when  great  accuracy  is  required,  this  value  must  be 
further  corrected  by  the  addition  of  the  value  of  Part  SECOND,  on 

ory-i          i^.  T   i  i     /cos-  0  c°s.  A2 
page  371,  multiplied  by  I :- J  cot.  z. 

If  the  chronometer  does  not  go  accurately  during  the  observa- 


EXPLANATION   OF  THE    TABLES  475 

tions,  a  further  correction  is  required  for  rate.  The  last  column 
af  page  371  furnishes  the  logarithm  of  this  correction  for  a  daily 
loss  or  gain  of  the  chronometer,  varying  from  0  to  30  seconds. 

An  example  of  the  use  of  this  Table  will  be  found  on  pages 
143-5. 

TABLE  XL,  pages  372-3,  is  for  determining  the  equation  of 
equal  altitudes  of  the  sun.  If  the  sun's  declination  remained 
the  same  from  the  forenoon  to  the  afternoon  observations,  it  is 
evident  that  half  this  interval,  added  to  the  time  of  the  first  ob- 
servation, would  give  the  time  of  apparent  noon  as  shown  by 
the  chronometer.  But  as  the  sun's  declination  is  continually 
changing,  a  correction  must  be  applied  on  account  of  this  varia- 
tion. This  correction  is  called  the  equation  to  equal  altitudes^ 
and  may  be  reduced  to  the  form 

x——  A.jw.  tang.  (/>  +  B  . fj, .  tang.  6. 

T 

Page  372  furnishes  the  value  of  — — : —  ,  which  is  repre- 
sented by  A.  This  must  be  multiplied  by  the  hourly  variation 
of  the  sun's  declination  (considered  as  negative  when  the  sun  is 
proceeding  toward  the  south),  and  also  by  the  tangent  of  the  lat- 

T 

itude  of  the  place.     Page  373  furnishes  the  value  of  — r, 

oU  tan.  7  g  _L 

which  is  represented  by  B.  This  must  be  multiplied  by  the 
hourly  variation  of  the  sun's  decimation,  and  also  by  the  tangent 
of  the  sun's  declination  at  the  time  of  apparent  noon  on  the  given 
day.  The  sum  of  these  two  quantities,  taken  with  their  proper 
signs,  is  the  correction  required. 

An  example  of  the  use  of  this  Table  will  be  found  on  page  129. 

TABLE  XII.,  pages  374-7,  furnishes  the  angle  of  the  vertical, 
the  logarithm  of  the  earth's  radius,  and  the  length  of  a  degree 
of  the  meridian,  and  also  of  a  parallel  of  latitude  for  every  de- 
gree of  latitude  from  the  equator  to  the  pole.  In  computing  the 
parallax  of  the  moon,  we  must  employ  the  geocentric  latitude, 
which  is  equal  to  the  observed  latitude  minus  the  angle  of  the 
vertical ;  and  we  must  also  employ  the  horizontal  parallax  be- 
longing to  the  place,  which  may  be  found  by  adding  the  loga- 
rithm of  the  horizontal  parallax  at  the  equator  to  the  logarithm 


476  PRACTICAL    ASTRONOMY. 

of  the  earth's  radius,  which  in  Table  XII.  is  set  against  the  given 
latitude.  The  use  of  these  numbers  is  explained  on  pages  184 
and  185. 

The  length  of  a  degree  of  the  meridian  and  of  a  parallel  is 
constantly  needed  in  Geodesy,  and  will  be  frequently  found  use- 
ful to  the  astronomer.  This  Table  is  taken  from  the  Berlin 
Jahrbuch  for  1852  ;  but  the  length  of  a  degree  of  longitude  and 
latitude,  which  is  there  given  in  toises,  has  been  carefully  con- 
verted into  English  feet,  and  it  is  hoped  will  be  found  correct  to 
the  last  decimal  place. 

TABLE  XIII.,  page  378,  shows  the  augmentation  of  the  moon's 
semi-diameter  on  account  of  her  apparent  altitude,  computed 
from  the  formula  page  200,  where  its  use  has  also  been  explained. 

TABLE  XIV.,  page  378,  shows  the  quantity  by  which  the 
moon's  equatorial  horizontal  parallax  must  be  diminished  to  ob- 
tain the  horizontal  parallax  belonging  to  any  other  latitude. 
This  reduction  is  given  for  three  values  of  the  moon's  equatorial 
parallax,  viz.,  53X,  57',  and  61X ;  and  for  any  other  value,  the 
equatorial  parallax  may  be  easily  found  by  interpolation.  The 
use  of  this  Table  will  be  understood  from  Art.  210,  page  186. 

TABLE  XV.,  page  379,  furnishes  the  parallax  of  the  sun  and 
planets  for  all  altitudes  above  the  horizon.  The  horizontal  par- 
allax is  to  be  sought  for  at  the  top  of  the  page,  and  the  altitude 
on  either  the  right  or  left  margin.  If  the  given  horizontal  par- 
allax is  not  found  exactly  in  the  Table,  the  parallax  in  altitude 
may  be  obtained  by  interpolating  between  the  numbers  given  in 
the  Table. 

Since  parallax  always  tends  to  diminish  the  true  altitude  of  a 
body,  we  must  add  the  parallax  to  the  observed  altitude  in  or- 
der to  obtain  the  true  altitude,  or  we  must  deduct  it  from  the 
observed  zenith  distance  in  order  to  obtain  the  true  zenith  dis- 
tance. 

TABLE  XVI.,  pages  380-3,  furnishes  the  moon's  parallax  in 
right  ascension,  and  also  in  declination  for  Cambridge  Observa- 
tory. The  form  of  the  Table  is  somewhat  complicated,  as  it  re- 


EXPLANATION   OF  THE    TABLES.  477 

quires  three  independent  arguments,  viz.,  the  moon's  declination, 
horizontal  parallax,  and  hour  angle.  The  Table  is  computed  for 
a  declination  of  0°,  5°,  10°,  15°,  20°,  and  25°  ;  for  a  horizon- 
tal  parallax  of  53X,  57X,  and  61' ;  and  for  every  five  or  ten  min- 
utes of  hour  angle  from  the  meridian.  For  any  value  of  these 
quantities  not  contained  in  the  Table,  a  double  or  triple  interpo- 
lation may  be  required.  If,  however,  we  neglect  the  second  dif- 
ferences, this  interpolation  may  be  readily  performed  as  follows : 
Find  what  number  in  the  Table  corresponds  most  nearly  to  the 
given  declination,  horizontal  parallax,  and  hour  angle.  Call  this 
number  the  approximate  parallax,  and  compute  the  correction 
which  should  be  added  or  subtracted  on  account  of  the  variation 
of  the  given  arguments  from  the  arguments  of  the  Table.  This 
method  will  be  understood  from  the  following  example. 

Example.  Required  the  moon's  parallax  in  right  ascension 
and  declination  for  Cambridge  Observatory,  when  the  moon's 
declination  is  19°  58'  9/x.4  N.,  the  horizontal  parallax  of  the 
place  60'  14x/.7,  and  the  moon's  hour  angle  3h.  47m.  23.50s. 

For  Right  Ascension. 

The  nearest  decimation  in  the  Table  is  20°  ;  the  nearest  hori- 
zontal parallax  is  6 17 ;  and  the  nearest  hour  angle  is  230m.  The 
corresponding  parallax  on  page  381  is  163.47s.  The  given  hour 
angle  is  less  than  230m.  by  2.61m.  To  find  the  correction  for 
2.61m.,  we  form  the  proportion 

10m. :  2.61m. ::  4.62s. :  1.20s. 

The  given  horizontal  parallax  is  less  than  61'  by  45x/.3.  To 
find  the  correction  for  45/x.3,  we  form  the  proportion 

240/x :  45x/.3 ::  10.7%. :  2.03s. 

The  given  declination  is  less  than  20°  by  110".6.  To  find 
the  correction  for  110/x.6,  we  form  the  proportion 

5°  or  18000"  :  110",6  ::  4.47s. :  0.02s. 
The  required  parallax  will  therefore  be 
163.47s. 
-1.20s. 
-2.03s. 
-   .02s. 

160.22s.,  which  corresponds  well  with 
the  result  on  page  250. 


478  PRACTICAL    ASTRONOMY. 

* 

For  Declination. 

The  approximate  parallax,  found  in  a  similar  manner  from 
page  383,  is  1802/x.O,  which  corresponds  to  Dec.  20°  N. ;  hori- 
zontal parallax,  61' ;  and  hour  angle,  220m.  The  corrections 
for  variation  of  the  proposed  arguments  from  the  preceding  ar- 
guments are  found  by  the  proportions 

20m. :  7.39m. ::    67/x.8  :  25/x.O,  the  correction  for  hour  angle. 

240//:45//.3   ::  119x/.7  :22x/.6,  the  correction  for  horizontal 

parallax. 
18000/x :  110/x.6 ::  193/x.3  :    1".2,  the  correction  for  decimation. 

The  required  parallax  will  therefore  be 

1802".0 + 25X/.0  -  22x/.6  +  l/x.2  =  1805".6, 
which  differs  less  than  a  second  from  the  result  on  page  250, 
and  this  discrepancy  arises  from  our  having  neglected  second 
differences  in  interpolation. 

When  it  is  required  to  compute  a  long  series  of  occupations 
and  eclipses  for  a  particular  place,  it  is  convenient  to  have  a  ta- 
ble of  parallaxes  like  the  preceding,  and  then  the  subsequent 
computation  occupies  but  a  few  minutes.  With  but  little  addi- 
tional labor,  this  Table  might  be  very  much  expanded,  so  that 
the  parallaxes  for  any  arguments  might  be  taken  out  by  mere 
inspection. 

TABLE  XVIL,  page  384,  contains  the  angles  formed  by  the 
intersection  of  a  vertical  circle  and  hour  circle  for  every  degree 
of  declination  from  29°  north  to  29°  south,  and  for  every  ten 
degrees  of  hour  angle  from  the  meridian  to  the  horizon.  A 
knowledge  of  these  angles  is  convenient  in  all  observations  of  the 
moon,  out  of  the  meridfan,  but  especially  in  observing  eclipses 
and  occultations.  The  method  of  computing  this  Table  has  been 
explained  in  Art.  145.  Every  astronomer  will  find  it  conven- 
ient to  compute  a  similar  table  for  his  own  observatory. 

TABLE  XYIIL,  page  385,  shows  the  correction  to  be  added  to 
the  moon's  declination  in  computing  an  occupation  qr  eclipse. 
The  reason  of  this  correction  has  been  explained  in  Art.  228. 
The  declination  of  the  moon  is  given  to  every  half  degree  in  the 
first  column,  and  the  difference  of  right  ascension  between  the 


EXPLANATION   OF   THE    TABLES.  479 

moon  and  star  in  the  case  of  an  occupation,  or  between  the  moon 
and  sun  in  the  case  of  a  solar  eclipse,  is  given  at  the  top  of  the 
page,  to  every  five  minutes  of  arc. 

TABLE  XIX.,  pages  386-7,  shows  the  semi-diurnal  arc,  or  the 
interval  of  time  employed  by  the  sun  or  a  star  in  passing  from 
the  horizon  to  its  point  of  culmination,  and  vice  versa,  accord- 
ing to  its  declination  and  the  latitude  of  the  place.  These  val- 
ues have  been  computed  from  formula  (2),  page  114,  without 
considering  the  effect  of  refraction,  which  would  increase  the 
duration  two  or  three  minutes,  and  sometimes  more  than  this. 
The  latitude  of  the  place  is  given  at  the  top  of  the  page,  and  the 
declination  of  the  star  in  the  first  vertical  column.  The  num- 
bers in  this  Table  are  to  be  subtracted  from  the  time  of  meridian 
passage  for  rising,  and  added  to  the  time  of  meridian  passage 
for  setting,  as  in  the  following  examples : 

Example  1.  Required  the  mean  time  of  setting  of  the  planet 
Venus,  July  5,  1855,  at  New  Haven,  lat.  41°  187,  the  declina- 
tion of  the  planet  being  13°  30/  N. 

Meridian  passage  July  5,  by  Nautical  Almanac  .  .  .  3h.  9m. 
Semi-diurnal  arc  for  lat.  41°  18X,  and  dec.  13°  3(K  N. 

(page  387) 6h.  49m. 

Venus  sets  July  5th,  1855 9h.  58m7 

Example  2.  Required  the  mean  time  of  rising  of  the  planet 
Jupiter,  July  5,  1855,  at  New  Haven,  the  declination  of  the 
planet  being  11°  40/  S. 

Meridian  passage  July  5,  by  Nautical  Almanac  .  .  .  15h.  24m. 
Semi-diurnal  arc  for  lat.  41°  18',  and  dec.  11°  40X  S. 

(page  387) .     5h.  18m. 

Jupiter  rises  July  5th,  1855    .........  lOh.    Gm~ 

This  Table  is  designed  for  northern  latitudes,  but  it  is  equally 
applicable  to  southern  latitudes  by  changing  the  declination  of 
the  star  from  N.  to  S.,  and  vice  versa. 

TABLE  XX.,  page  388,  contains  a  comparison  of  French  mil- 
limeters with  English  inches,  and  will  be  found  convenient  for 
reducing  French  measures  into  English.  It  is  deduced  from  the 
assumption  that  the  French  metre  at  the  freezing  point  is  equal 
to  39.37079  English  inches  at  the  temperature  of  62°  Fahren- 
heit ;  the  standard  temperature  of  the  French  scale  being  32° 


480  PRACTICAL   ASTRONOMY. 

Fahrenheit,  and  that  of  the  English  scale  being  62°  Fahrenheit. 
This  is  the  result  given  by  Captain  Kater  in  the  Philosophical 
Transactions  for  1818,  page  109.  The  table  of  proportional  parts 
in  the  last  column  gives  the  value  of  tenths  of  a  millimeter  in 
English  inches,  and  will  serve  for  hundredths  by  removing  the 
decimal  point  one  place  to  the  left. 

The  relation  of  the  metre  to  the  yard  adopted  by  the  "United 
States  Coast  Survey  is, 

1  metre  =  1.0935696  yards,  or  39.3685  United  States  standard 
inches. 

TABLE  XXL,  page  389,  enables  us  to  convert  English  inches 
into  millimeters,  and  is  derived  from  the  same  data  as  the  pre- 
ceding Table.  The  table  of  proportional  parts  in  the  last  column 
gives  the  values  of  hundredths  of  an  inch  in  millimeters,  and  will 
serve  for  thousandths  by  removing  the  decimal  point  one  place 
to  the  left. 

TABLE  XXII. ,  pages  390-1,  is  designed  for  computing  the  dif- 
ference in  the  heights  of  two  places  by  means  of  the  barometer. 
This  Table  was  computed  from  the  formula  of  Laplace,  modi- 
fied in  accordance  with  the  results  of  more  recent  determinations. 
Suppose  that  we  have  observed 

(  H,  the  height  of  the  barometer. 

At  the  lower  station,  <  T,    the  temperature  of  the  barometer. 
(  £,     the  temperature  of  the  air. 
(  h',  the  height  of  the  barometer. 
At  the  upper  station,  <  T',  the  temperature  of  the  barometer. 

(  t',    the  temperature  of  the  air. 

Represent  by  s  the  height  of  the  lower  station  above  the  level 
of  the  sea,  by  L  the  latitude  of  the  place,  and  by  h  the  observed 
height,  h',  reduced  to  the  temperature  T. 

The  difference  of  level,  #,  between  the  two  stations  is  given 
by  the  formula : 


x= 60158.6  ft.  log.      x 


/     ;+*:-64\ 
V       900   ) 

(1 +  0.00265  cos.  2L) 
/'      x  +  52251  s 

I    •*•     I    n  /S  o  C)  c^  r*  c\  c\  ""  * 


IV   20888629  '  10444315 


EXPLANATION   OF  THE    TABLES.  481 

But  h  represents  the  height  /*',  reduced  from  the  temperature 
Tx  to  the  temperature  T.  The  expansion  of  mercury  for  1° 
Fahrenheit  is  0.0001000  ;  that  of  the  brass,  which  forms  the 
scale  of  the  barometer,  is  0.0000104  ;  the  difference  is  0.0000896. 
Hence  we  have  A=  A'  {1  +  0.0000896(1-^')}. 

Therefore, 

60158.6  ft.  log.  5=60158.6  ft.  log.  ^~2M09  ft.  (T-T'). 
/i  li 

Part  I.  of  the  Table  furnishes  in  English  feet  the  value  of  the 
expression  60158.6  log.  H  for  heights  of  the  barometer  from  11 
to  31  inches  ;  only  they  have  all  been  diminished  by  the  con- 
stant 27541.5  feet,  which  does  not  change  the  difference, 
60158.6  log  H-  60158.6  log  /*. 

Part  II.  furnishes  the  correction  —  2.3409(T  —  Tx),  depending 
upon  the  difference  T  —  Tx  of  the  temperatures  of  the  barometers 
at  the  two  stations.  This  correction  is  generally  negative.  It 
would  be  positive  if  T  —  Tx  were  negative  ;  that  is,  if  the  tem- 
perature Tx  of  the  barometer  at  the  upper  station  exceeded  the 
temperature  T  at  the  lower  station. 

Part  III.  gives  the  correction  Ax  0.00265  cos.  2L,  to  be  ap- 
plied to  the  approximate  altitude  A,  and  which  arises  from  the 
variation  of  gravity  from  the  latitude  of  45  degrees  to  the  lati- 
tude L  of  the  place  of  observation.  This  correction  has  the  same 
sign  as  cos.  2L  ;  that  is,  it  is  positive  from  the  equator  to  45  de- 
grees, and  negative  from  45  degrees  to  the  pole. 

Part  IV.  gives  the  correction  A  .  —  ^   ^?r>  which  is  always 


to  be  added  to  the  approximate  height  A,  and  which  is  due  to 
the  diminution  of  gravity  on  the  vertical. 

Part  V.  furnishes  for  the  approximate  difference  of  level,  A,  the 

g 
small  correction  A  .  ,  corresponding  to  several  values 

J_  vJ  J.  i.  A.  O  -L  O 

of  the  height  s  of  the  lower  station.  But  in  place  of  s  there 
has  been  substituted,  as  the  argument  of  the  table,  the  height 
H  of  the  barometer  at  this  station. 

Method  of  Computation. 

Take  from  Part  L,  page  390,  -the  two  numbers  corresponding 
to  the  observed  barometric  heights  H  and  h'.     From  their  differ- 

HII 


482  PRACTICAL   ASTRONOMY. 

ence  subtract  the  correction  2.3409(T  — Tx),  found  in  Part  II., 
with  the  difference  T  —  Tx  of  the  thermometers  attached  to  the 
barometers.  "We  thus  obtain  an  approximate  altitude,  a. 


"We  then  calculate  the  correction  a .  — —^r^ —  for  the  temper- 
ature of  the  air,  by  multiplying  the  nine  hundredth  part  of  a  by 
the  sum  of  the  temperatures  t  and  f,  diminished  by  64.  This 
correction  is  of  the  same  sign  as  £-Hx  — 64.  "We  thus  obtain  a 
second  approximate  altitude,  A. 

"With  A  and  the  latitude  of  the  place,  L,  we  seek,  in  Part  III., 
the  correction  A  x  0.00265  cos.  2L,  arising  from  the  variation  of 
gravity  with  the  latitude. 

For  the  approximate  height  A,  Part  IV.  gives  the  correction 

A+52251      .  .       ,.       ,,     ,.    .      ,.        . 
A  x  onQQQ7^>  arising  from  the  diminution  of  gravity  on  a  vert- 


ical.     This  correction  is  always  additive. 

Finally,  when  the  height,  s,  of  the  lower  station  is  considera- 

o 

ble,  the  small  correction  A  x          /icTg  may  ^e  f°und  in  Part  V. 

This  correction  is  always  additive. 

Example  1.  M.  Humboldt  made  the  following  observations 
on  the  mountain  of  Gruanaxuato,  in  Mexico,  in  latitude  21°,  viz. : 

TTnnpr  Station  Lower  Station, 

lon'  On  the  Bank  of  the  Sea. 

Thermometer  in  open  air    ...  t'  —70.3  t  =77.5 

'Thermometer  to  barometer   .  .  T'^70.3  T^77.5 

Barometer A' =23.660         H= 30.046 

What  was  the  difference  in  the  height  of  the  two  stations  ? 

(for  H  =  30.046  inches  27649.7 

S  (  for  h'  =  23.660  inches  21406.9 

Difference  .  .  .  6242.8 

Part  II.  gives  for  T  -  Tx  =  7.2°  -16.9 

Approximate  altitude,  a 6225.9 

JLjt +t'-  64)  =6.918  x  83.8^  +  579.7 

you 

Second  approximate  altitude,  A  .  .  6805.6 
Part  IE.  gives  for  A=6806,  and  L  =  21°  +  13.3 
Part  IY.  gives  for  6806  +  19.3 

Height  above  the  sea 6838.2  feet. 


EXPLANATION   OF   THE    TABLES.  483 

Example  2.  M.  Gray  Lussac,  in  his  celebrated  balloon  ascent 
in  1805,  found  his  barometer  to  indicate  12.945  English  inches, 
the  temperature  being  14.9°  Fahrenheit.  The  barometer  at 
Paris  at  the  same  time  indicated  30.145  English  inches,  with  a 
temperature  of  87.44°  Fahrenheit.  Required  the  elevation  of 
the  balloon  above  Paris. 

(for  H:=  30.145  inches  27735.6 

18  I  for  A'  -  12.945  inches  5650.4 

Difference  .  .  .  22085.2 

Part  II.  gives  for  T  -  T  =  72.54°  -  169.9 

Approximate  altitude,  a  .....  21915.3 

(*  +  1'  -  64)  =  24.35  x  38.34°  +  933.6 


Second  approximate  altitude,  A  .  22848.9 
Part  III.  gives  for  A=22848,  and  L  =48°  50'  -  8.2 
Part  IV.  gives  for  22848  +82.1 

Height  of  balloon  above  Paris  .  .  22922.8  feet. 

TABLE  XXIII.  ,  pages  392-3,  furnishes  the  coefficients  for  in- 
terpolation by  differences.  The  Table  on  page  392  contains  the 
values  of  the  coefficients  for  interpolation  by  Bessel's  formula, 
given  in  Art.  223.  Column  first  contains  the  values  of  t  to  each 
hundredth  of  unity.  Column  second  contains  the  values  of  the 

factor  t  .—   —  for  each  value  of  t  contained  in  the  first  column. 

m 

Column  third  contains  the  values  of  the  factor  v  —  *•)(*—?)  for 

2  .  3 

each  value  of  t  contained  in  the  first  column.     Column  fourth 

contains  the  values  of  the  factor  5  —  _  /  V  ~   '?  and  column 

-£.0.4 


fifth  contains  the  values  of  the  factor  (*+1X(*~1)  (*--3)  (*-•*) 

2.3.4.5 
for  each  value  of  t  contained  in  the  first  column. 

The  coefficients  of  the  second  differences  are  negative  ;  the 
coefficients  of  the  third  differences  are  positive  for  values  of  t 
less  than  one  half,  and  negative  for  values  of  t  greater  than  one 
half.  The  coefficients  of  the  fourth  differences  are  invariably 
positive  ;  the  coefficients  of  the  fifth  differences  are  negative  for 
values  of  t  less  than  one  half,  and  positive  for  values  of  t  greater 


484  PRACTICAL   ASTRONOMY. 

than  one  half.  The  mode  of  using  this  Table  has  been  explain- 
ed in  Art.  223. 

The  Table  on  page  393  contains  the  values  of  the  coefficients 

t(t-l)  t(t-l)(t-2)  t(t—l)(t-2)(t—3) 
t,  ^-4  -      33     -,  J         2  "34  '       '  '        g 

the  same  as  the  coefficients  of  the  binomial  formula,  are  called 
binomial  coefficients,  to  distinguish  them  from  Bessel's  coeffi- 
cients on  page  392.  Columns  first  and  second  are  the  same  as 
on  page  392.  Column  third  contains  the  values  of  the  factor 

*  ~      0~      for  each  value  of  t  contained  in  the  first  column, 
2  .  o 

and  the  subsequent  columns  are  constructed  in  a  similar  man- 
ner. The  coefficients  for  the  odd  differences  are  positive,  while 
those  for  the  even  differences  are  negative.  The  mode  of  using 
this  Table  has  been  explained  in  Art.  220. 

TABLE  XXIV.,  pages  394-6,  contains  the  logarithms  of  the 
coefficients  for  interpolation  by  Bessel's  formula  for  every  five 
minutes,  the  unit  of  time  being  supposed  to  be  12  hours.  This 
Table  is  from  Sawitsch's  Practischen  Astronomic,  and  its  use 
has  been  explained  on  page  208. 

TABLE  XXV.,  page  397,  enables  us  to  convert  degrees  of  the 
centesimal  thermometer  into  degrees  of  Fahrenheit.  It  is  found- 

9 
ed  on  the  equation,  x°  centesimal  =  (32°  +  -#°)  Fahrenheit. 


TABLE  XXVL,  page  397,  enables  us  to  convert  degrees  of 
Reaumur's  thermometer  into  degrees  of  Fahrenheit.    It  is  found- 

9 
ed  on  the  equation,  £°  Reaumur  =  (32  °-f-x°)  Fahrenheit. 


TABLE  XXVIL,  page  398,  shows  the  height  of  the  barometer 
corresponding  to  temperatures  of  boiling  water  from  185°  to  214° 
Fahrenheit.  The  temperature  at  which  water  boils  in  the  open 
air  depends  upon  the  weight  of  the  atmospheric  column  above 
it,  and  under  a  diminished  barometric  pressure  the  water  will 
boil  at  a  lower  temperature.  Since  the  weight  of  the  atmos- 
phere decreases  with  the  elevation,  it  is  evident  that,  in  ascend- 


EXPLANATION   OF   THE    TABLES.  485 

ing  a  mountain,  the  higher  the  station  the  lower  will  be  the 
temperature  at  which  water  boils.  Hence,  if  we  knew  the  height 
of  the  barometer  corresponding  to  the  temperature  of  boiling  wa- 
ter, we  could  measure  the  altitude  of  a  mountain  by  observing 
the  temperature  at  which  water  boils.  Table  XXVII.  is  de- 
rived from  a  Table  by  Regnault,  published  in  the  Annales  de 
Physique  et  de  Chimie,  t.  xiv.,  p.  206.  In  RegnauhVs  Table 
the  temperature  is  expressed  in  centigrade  degrees,  and  the 
height  of  the  barometer  in  millimeters.  I  have  deduced  from 
this  a  new  Table,  in  which  the  temperature  is  expressed  in  de- 
grees of  Fahrenheit,  and  the  height  of  the  barometer  in  English 
inches. 

TABLE  XXVIII. ,  page  399,  contains  the  depression  of  mercury 
in  glass  tubes  on  account  of  capillarity,  according  to  several  dif- 
ferent authorities.  i  £tfi* 
io-  j*l 

TABLE  XXIX.,  page  399,  contains  the  factors  by  which  the 
difference  of  readings  of  the  dry-bulb  and  wet-bulb  thermome- 
ters must  be  multiplied  in  order  to  produce  the  difference  be- 
tween the  readings  of  the  dry-bulb  and  dew-point  thermome- 
ters. These  factors  are  derived  from  a  long  series  of  observa- 
tions made  at  the  Greenwich  Observatory,  and  enable  us  to  con- 
vert observations  made  with  the  wet-bulb  thermometer  into  ob- 
servations made  with  Daniell's  hygrometer. 

Example  1.  The  temperature  of  the  air  being  81.3°,  and  that 
of  the  wet-bulb  being  68.9°,  it  is  required  to  determine  the  dew- 
point. 

The  difference  between  the  dry  and  wet  bulb  thermometers 
is  12.4°,  which,  multiplied  by  1.5,  gives  18.6°,  which  is  the 
difference  between  the  dry-bulb  and  dew-point  thermometers. 
Hence  the  dew-point  was  at  62.7°. 

Example  2.  The  temperature  of  the  air  being  46.9°,  and  that 
of  the  wet-bulb  thermometer  44.2°,  it  is  required  to  determine 
the  dew-point. 

The  difference  between  the  dry  and  wet  bulb  thermometers 
is  2.7°,  which,  multiplied  by  2.2,  gives  5.9°.  Hence  the  dew- 
point  was  at  41.0°. 


486  PRACTICAL   ASTRONOMY. 

TABLE  XXX.,  pages  400-459,  is  a  Catalogue  of  1500  stars, 
derived  chiefly  from  the  Catalogue  of  the  British  Association. 
This  Catalogue  contains  all  the  stars  of  the  British  Association 
Catalogue  to  the  fifth  magnitude  inclusive,  and  about  a  dozen 
stars  of  the  magnitude  five  and  a  half,  situated  within  a  few 
degrees  of  the  north  pole. 

Column  first,  on  the  left-hand  page,  contains  the  number  of 
the  star  in  this  Catalogue ;  column  second  contains  the  equiva- 
lent number  in  the  Catalogue  of  the  British  Association ;  column 
third  contains  the  name  of  the  constellation  to  which  the  star 
belongs,  together  with  Flamsteed's  numbers  and  Bayer's  letters, 
according  to  the  British  Association  Catalogue  ;  column  fourth 
contains  the  magnitude  of  the  star  according  to  the  same  Cata- 
logue ;  column  fifth  contains  its  right  ascension  on  the  1st  of 
January,  1850 ;  column  sixth  contains  the  annual  variation  of 
the  right  ascension,  and  includes  proper  motion  where  it  exists ; 
column  seventh  contains  the  north  polar  distance  on  the  1st  of 
January,  1850 ;  and  column  eighth  contains  the  annual  varia- 
tion of  polar  distance,  including  proper  motion. 

On  the  right-hand  page,  column  first  contains  the  number  of 
the  star  repeated  from  the  former  page ;  the  next  four  columns 
contain  the  logarithms  of  the  factors  a,  &,  c,  and  d,  for  computing 
the  reduction  from  the  mean  to  the  apparent  right  ascension ; 
while  the  last  four  columns  contain  the  logarithms  of  the  factors 
a',  b',  c',  and  d',  for  computing  the  reduction  from  the  mean  to 
the  apparent  polar  distance.  All  the  numbers  on  each  page  are 
copied  from  the  British  Association  Catalogue,  with  the  excep- 
tion of  the  right  ascensions  and  polar  distances  of  such  of  the 
stars  as  are  contained  in  the  Greenwich  twelve-year  Catalogue. 
The  places  of  such  stars  have  been  carefully  reduced  from  the 
years  1840  and  1845  to  1850,  and  are  distinguished  from  other 
numbers  in  the  same  columns  by  an  asterisk. 

In  a  few  cases,  in  which  the  Greenwich  twelve-year  Cata- 
logue differs  considerably  from  the  British  Association  Catalogue, 
the  results  of  the  most  recent  observations  at  Greenwich  have 
been  combined  with  former  ones,  to  obtain  the  mean  places 
which  are  incorporated  in  this  Table. 

The  mode  of  deducing  the  apparent  places  of  the  stars  from 
their  mean  places  has  been  explained  on  page  220. 


EXPLANATION    OP   THE    TABLES.  487 

TABLE  XXXI.  ,  page  460,  contains  the  secular  variation  of  the 
annual  precession  in  right  ascension  for  the  stars  of  Table  XXX. 
whenever  this  variation  exceeds  0.035s.  The  annual  precession 
of  a  star  does  not  remain  the  same  for  a  long  period  of  time,  but 
undergoes  a  slight  increase  or  decrease  from  year  to  year.  As 
this  annual  change  of  the  precession  is  generally  small  in  amount, 
and  constant  for  a  very  long  period,  it  is  commonly  known  by 
the  name  of  the  secular  variation  ;  for,  when  inserted  in  ta- 
bles, as  on  page  460,  it  is  usually  multiplied  by  100,  for  the 
sake  of  a  convenient  arrangement  of  the  figures. 

Assuming,  therefore,  the  annual  variation  of  a  star  in  the 
Catalogue  to  be  denoted  by  Y  (which  is  equal  to  the  sum  of  the 
annual  precession  and  the  proper  motion),  the  secular  variation 
by  S,  the  change  of  position  in  the  star  (either  in  right  ascension 
or  north  polar  distance,  as  the  case  may  be)  on  January  1st 
(1850  +  #),  will  be  expressed  by 

S 


where  #,  which  denotes  the  number  of  years  from  1850,  must 
be  assumed  +  after,  and  —  before,  that  epoch.  And  in  this 
manner  the  mean  place  of  a  star  should  be  brought  up  from 
the  epoch  1850  to  the  commencement  of  any  other  required  year 
before  we  apply  the  annual  correction  for  precession,  aberration, 
and  nutation.  But  for  most  stars,  when  the  period  is  not  very 
long,  the  secular  variation  may  be  omitted. 

Example.  It  is  required  to  find  the  mean  right  ascension  of 
star  46,  on  page  400,  for  January  1,  1860. 


Here  y  =  10  years.     Hence  =  J. 

100 

From  page  460,  S=+  1.2222s.,  and  |=+  0.61s. 

Hence  we  have  the  following  results  : 

Mean  right  ascension  January  1,  1850    .  Oh.  49m.    9.55s. 

Variation  in  10  years  ...........        -flrn.    7.16s. 

Correction  for  secular  variation    .....  -f-  0.61s. 

Mean  right  ascension  January  1,  1860    .  Oh.  50m.  17.32s. 

TABLE  XXXIL,  page  461,  contains  the  secular  variation  of  the 
annual  precession  in  north  polar  distance  for  all  stars  in  Table 


488  PRACTICAL    ASTRONOMY. 

XXX.  whenever  this  variation  amounts  to  Oxx.43.     This  Table 
is  to  be  used  in  the  same  manner  as  the  preceding. 

Example.  It  is  required  to  find  the  mean  north  polar  distance 
of  star  300,  page  410,  for  January  1,  1860. 


, 

From  page  461,  S=  +  1".367,  and  |=  +  0XX.7. 

Hence  we  have  the  following  results  : 

Mean  north  polar  distance  January  1,  1850  .  10°  57X  24XX.2 

Variation  in  10  years  .............  —  54/x.2 

Correction  for  secular  variation  ........  4-   Ox/.7 

Mean  north  polar  distance  January  1,  1860  .  10°  56X  30x/.7 
For  most  of  the  stars,  the  secular  variation  of  precession  is  in- 
appreciable, except  for  long  intervals  of  time. 

TABLE  XXXIII.,  page  462,  contains  the  principal  elements  of 
•the  planetary  system,  taken  chiefly  from  Madler's  Populare  As- 
tronomie,  vierte  Auflage.  I  have  substituted  the  English  de- 
nominations for  measures  of  length,  in  place  of  the  foreign  de- 
nominations of  Madler,  and  have  substituted  more  recent  ele- 
ments of  Neptune.  Several  of  the  numbers  in  Madler's  Table 
have  been  changed  in  accordance  with  what  were  considered  to 
be  the  best  authorities. 

TABLE  XXXIV.,  page  463,  contains  the  elements  of  the  satel- 
lites of  the  primary  planets.  The  elements  of  the  moon  were 
derived  from  "Bailey's  Astronomical  Tables  and  Formulae." 
Those  of  Jupiter's  satellites  were  derived  from  Madler's  Astro- 
nomie  ;  those  of  Saturn's  satellites  were  derived  chiefly  from 
Madler,  modified  in  some  instances  by  comparison  with  Her- 
schel's  Astronomy  and  Hind's  Solar  System.  The  elements  of 
the  satelites  of  Uranus  were  derived  by  myself  chiefly  from  the 
observations  of  Lassell  ;  and  those  of  the  satellite  of  Neptune 
were  derived  from  Hind's  Solar  System. 

TABLE  XXXV.,  pages  464-5,  contains  the  elements  of  the  as- 
teroids. These  elements  have  all  been  derived  from  the  Berlin 
Astronomisches  Jahrbuch  for  1855,  with  the  following  excep- 


EXPLANATION    OF    THE    TABLES.  489 

tions,  viz.,  Nos.  24,  27,  28,  29,  30,  31,  32,  and  33,  which  were 
taken  from  recent  pages  of  the  Astronomische  Nachrichten ;  and 
No.  5  from  the  English  Nautical  Almanac  for  1854. 

TABLE  XXXVI.,  pages  466-7,  furnishes  the  constants  for  ob- 
taining with  the  greatest  accuracy  the  sines  and  tangents  of 
arcs  not  exceeding  two  degrees.  The  column  headed  log.  sin. 
A.  — log,  Axx  furnishes  the  difference  between  the  logarithmic 
sine  of  the  arc  given  in  the  adjacent  column,  and  the  logarithm 
of  that  arc  expressed  in  seconds.  Thus, 

The  logarithmic  sine  of  0°  40X  is 8.06577631 

The  logarithm  of  2400//(=40/)  is 3.38021124 

The  difference  is 4.68556507 

This  is  the  number  found  on  page  466,  under  the  heading  log. 
sin.  A— log.  Axx,  opposite  0°  40X;  and  in  a  similar  manner  the 
other  numbers  in  the  Table  were  obtained.  These  numbers 
vary  quite  slowly  for  two  degrees  ;  and  hence,  to  find  the  loga- 
rithmic sine  of  an  arc  not  exceeding  two  degrees,  we  have  but 
to  add  the  logarithm  of  the  arc  expressed  in  seconds  to  the  ap- 
propriate number  found  in  this  Table. 

Required  the  logarithmic  sine  of  0°  24X  22XX.57. 

Tabular  number  from  page  466 4.6855712 

The  logarithm  of  1462XX.57  is 3.1651167 

The  logarithmic  sine  of  0°  24X  22XX.57  is  ....  7^8506879 

The  logarithmic  tangent  of  an  arc  not  exceeding  two  degrees 
is  found  in  a  similar  manner 

The  same  Table  enables  us  to  find  the  arc  corresponding  to  a 
given  logarithmic  sine  or  tangent.  If  from  the  given  logarith- 
mic sine,  we  subtract  the  corresponding  tabular  number  on  page 
466,  the  remainder  will  be  the  logarithm  of  the  arc  expressed  in 
seconds. 

Required  the  arc  corresponding  to  the  logarithmic  sine 
7.0000000.  We  find  from  page  466  that  the  arc  must  be 
nearly  3X ;  the  corresponding  tabular  number  on  page  466  is 
4.6855748. 

The  difference  is  2.3144252, 

which  is  the  logarithm  of  206.265. 

Hence  the  required  arc  is  3X  26XX.265. 


490  PRACTICAL    ASTRONOMY. 

In  the  same  manner  we  may  find  the  are  corresponding  to  a 
logarithmic  tangent. 

The  numbers  in  Table  XXXVI.  are  given  to  8  decimal  places, 
in  order  that  we  may  be  sure  of  getting  the  seventh  figure  cor- 
rect to  the  nearest  decimal ;  it  will  be  of  no  use,  however,  to 
retain  the  eighth  figure  in  our  computations,  unless  we  emplov 
logarithmic  tables  of  more  than  seven  decimal  places. 

TABLE  XXX VII. ,  page  468,  contains  a  miscellaneous  collec- 
tion of  numbers  which  are  most  frequently  employed  in  compu- 
tations. They  are  derived  chiefly  from  Shortrede's  Logarithmic 
Tables. 


CATALOGUE  OF  ASTRONOMICAL  INSTRUMENTS  BY 
DIFFERENT  MAKERS,  WITH  THEIR  PRICES. 

Telescopes  by  Merz  and  Mahler. 

THE  refracting  telescopes  manufactured  at  the  establishment  of 
Merz  and  Mahler,  of  Munich,  have  acquired  a  higher  reputation 
than  any  others. 

The  following  is  the  list  of  instruments  furnished  by  this  estab- 
tablishment.  The  dimensions  are  given  in  French  measure,  ac- 
cording to  which,  one  inch  — 1.0658  English  inches  ;  and  one  foot— 
12.7892  English  inches.  The  prices  are  in  francs,  one  franc  being 
equal  to  18.5  cents. 

No.  1.  Achromatic  telescope  of  14  inches  aperture  and  21  feet 
focus,  with  an  hour  circle  17  inches  in  diameter,  divided  into  single 
seconds  of  time,  and  a  declination  circle  24  inches  in  diameter,  di- 
vided to  4  seconds  of  arc,  with  six  common  astronomical  eye-pieces, 
magnifying  140,  226,  336,  504,  756,  and  1200  times ;  and  nine  mi- 
crometric  eye-pieces,  magnifying  from  148  to  2000  times.  The  find- 
er has  a  focal  length  of  42  inches,  and  an  aperture  of  34  lines. 

Price  91,300  francs. 

This  telescope  is  of  the  same  size  as  those  of  Cambridge  and  Pul- 
kova. 

No.  2.  Achromatic  telescope  of  12  inches  aperture  and  17J  feet 
focus,  like  the  preceding  in  all  respects,  except  that  it  has  six  as- 
tronomical eye-pieces,  magnifying  116,  188,  280,  420,  630,  and  1000 
times ;  and  nine  micrometric  eye-pieces,  magnifying  from  124  to 
1200.  Price  65,220  francs. 

This  telescope  is  of  the  same  dimensions  as  that  belonging  to  the 
Cincinnati  Observatory. 

No.  3.  Achromatic  telescope  of  10^-  inches  aperture  and  15  feet 
focal  length,  with  an  hour  circle  15  inches  in  diameter,  divided  to 
two  seconds  of  time  ;  a  decimation  circle  similar  to  No.  1  ;  five  ordi- 
nary eye-pieces,  magnifying  160,  240,  360,  540,  and  856  times  ;  and 
eight  micrometric  eye-pieces,  magnifying  from  100  to  1200  times. 
The  finder  has  a  focal  length  of  30  inches,  and  an  aperture  of  29 
lines.  Price  47,830  francs. 


492  PRACTICAL   ASTRONOMY. 

No.  4.  Achromatic  telescope  of  9  inches  aperture  and  13J  feet 
focal  length,  with  an  hour  circle  of  14  inches  diameter,  divided  to 
two  seconds  of  time  ;  arid  a  declination  circle  of  20  inches  diameter, 
divided  to  four  seconds  ;  five  astronomical  eye-pieces,  magnifying 
142, 212,  320, 480,  and  760  times  ;  and  eight  micrometric  eye-pieces, 
magnifying  from  94  to  1000  times.  The  finder  the  same  as  for  No.  3. 

Price  32,610  francs. 

This  telescope  is  of  the  same  dimensions  as  that  belonging  to  the 
National  Observatory  at  Washington. 

No.  5.  Achromatic  telescope  of  7  inches  aperture,  9§  feet  focal 
length,  with  an  hour  circle  9^  inches  in  diameter,  divided  to  four 
seconds  of  time  ;  and  a  declination  circle  15  inches  in  diameter,  di- 
vided to  ten  seconds  ;  five  astronomical  eye-pieces,  magnifying  102, 
146,  232,  348,  and  550  times  ;  and  six  micrometric  eye-pieces,  mag- 
nifying from  100  to  580  times.  The  finder  has  a  focal  length  of  20 
inches,  and  an  aperture  of  21  lines.  Price  17,390  francs. 

This  telescope  is  of  the  same  dimensions  as  that  belonging  to 
Shelby  College,  Kentucky. 

No.  6.  Achromatic  telescope  of  6  inches  aperture  and  8  feet  focal 
length,  with  an  hour  circle  9  inches  in  diameter,  divided  to  four  sec- 
onds of  time  ;  and  a  declination  circle  12  inches  in  diameter,  divided 
to  ten  seconds ;  five  astronomical  eye-pieces,  magnifying  85,  127, 
192,  288,  and  456  times  ;  and  five  micrometric  eye-pieces,  magnify- 
ing from  128  to  480  times.  The  finder  has  an  aperture  of  19  lines, 
and  a  focal  length  of  20  inches.  Price  9240  francs. 

Telescopes  of  this  number  belong  to  the  High  School  Observa- 
tory at  Philadelphia ;  to  Sharon  Observatory,  Pennsylvania  ;  and  to 
Dartmouth  College. 

All  the  preceding  numbers  are  furnished  with  clock-work. 

No.  7.  Achromatic  telescope  of  52  lines  aperture  and  6  feet  focal 
length,  with  an  hour  circle  8  inches  in  diameter,  divided  to  four  sec- 
onds of  time  ;  and  a  declination  circle  10  inches  in  diameter,  divided 
to  ten  seconds  ;  five  astronomical  eye-pieces,  magnifying  from  64  to 
324  times  ;  a  terrestrial  eye-piece,  magnifying  82  times  ;  an  annular 
micrometer,  finder,  etc.,  without  clock-work.  Price  4782  francs. 

No.  8.  Achromatic  telescope  of  48  lines  aperture  and  5  feet  focus, 
with  five  astronomical  eye-pieces,  magnifying  from  54  to  270  times, 
and  a  terrestrial  eye-piece,  magnifying  66  times. 

Price  4348  francs. 

No.  9.  Achromatic  telescope  of  45  lines  aperture  and  4^  feet  fo- 
cus, on  a  brass  stand,  with  an  hour  circle  7  inches  in  diameter,  di- 
vided to  four  seconds  of  time  ;  and  a  declination  circle  7  inches  in 


CATALOGUE  OF  INSTRUMENTS.       493 

diameter,  divided  to  thirty  seconds ;  five  astronomical  eye-pieces, 
magnifying  from  48  to  243  times  ;  a  terrestrial  eye-piece,  magnify- 
ing 90  times ;  an  annular  micrometer,  finder,  etc. 

Price  3260  francs. 

No.  10.  Achromatic  telescope  of  37  lines  aperture  and  4  feet  fo- 
cus, with  magnifying  powers  from  64  to  216,  etc. 

Price  1956  francs. 
A  line  is  the  twelfth  part  of  an  inch. 

MERIDIAN  CIRCLES  AND  TRANSIT  INSTRUMENTS. 

Excellent  meridian  circles  and  transit  instruments  are  manufac- 
tured at  Munich,  Berlin,  Hamburg,  London,  and  Paris. 

Meridian  Circles,  by  Ertel  and  Son,  Munich. 

The  following  are  the  prices  of  meridian  circles  and  transit  in- 
struments manufactured  by  Ertel  and  Son,  of  Munich.  The  dimen- 
sions are  given  in  French  measure,  and  the  prices  in  francs,  as  on 
page  491. 

1.  Meridian  circle,  with  a  telescope  of  8  feet  focal  length  and  five 
and  a  half  inches  aperture.     At  one  extremity  of  the  horizontal  axis 
is  the  circle  of  altitude,  three  feet  and  four  inches  in  diameter,  read- 
ing, by  four  verniers,  to  one  second.     At  the  other  extremity  of  the 
axis  is  a  circle  of  the  same  dimensions,  but  divided  only  to  single 
minutes  by  one  vernier,  and  furnished  with  a  clamp  and  tangent 
screw.     The  instrument  has  a  large  level  and  four  astronomical 
eye-pieces.  Price  17,160  francs. 

Microscopes  may  be  substituted  for  the  verniers  without  any  in- 
crease of  price. 

2.  Meridian  circle,  with  a  telescope  of  5  feet  focus  and  51  lines 
aperture.     The  circle  is  three  feet  in  diameter,  and  is  divided  by 
four  verniers  to  two  seconds.  Price  11,580  francs. 

3.  Meridian  circle,  with  a  telescope  of  50  inches  focus  and  42  lines 
aperture.     The  circle  is  two  feet  in  diameter.     Price  6430  francs. 

4.  Meridian  circle,  with  a  telescope  of  42  inches  focus  and  34 
lines  aperture.     The  circle  is  twenty  inches  in  diameter,  and  di- 
vided by  four  verniers  to  four  seconds.  Price  5150  francs. 

5.  Transit  instrument,  with  an  object-glass  of  8  feet  focus  and  66 
lines  aperture,  and  four  astronomical  eye-pieces. 

Price  8150  francs. 

6.  Transit  instrument,  with  an  object-glass  of  6  feet  focus  and 
52  lines  aperture,  and  four  astronomical  eye-pieces. 

Price  5360  francs. 


494  PRACTICAL   ASTRONOMY. 

7.  Transit  instrument,  with  an  object-glass  of  42  inches  focus  and 
34  lines  aperture,  with  three  astronomical  eye-pieces. 

Price  2150  francs. 

8.  Portable  transit  instrument,  with  an  object-glass  of  22  inches 
focus  and  2  inches  aperture,  and  two  astronomical  eye-pieces.     It 
has  a  vertical  circle  of  6  inches,  divided  by  one  vernier  to  single 
minutes,  and  an  azimuth  circle  of  14  inches,  divided  by  four  ver- 
niers to  ten  seconds.  Price  1410  francs. 

9.  Portable  transit  instrument,  with  an  object-glass  of  18  inches 
focus  and  19  lines  aperture,  and  two  astronomical  eye-pieces.     It 
has  a  vertical  circle  of  5  inches,  and  an  azimuth  circle  of  12  inches. 

Price  1180  francs. 

10.  Grand  vertical  circle,  three  feet  and  four  inches  in  diameter, 
divided  to  two  minutes,  and  reading  by  four  microscopes.     The  tel- 
escope has  an  object-glass  of  6  feet  focus  and  5£  inches  aperture. 

Price  19,300  francs. 

11.  Vertical  repeating  circle,  three  feet  in  diameter,  divided  by 
four  verniers  to  two  seconds.     The  telescope  has  an  object-glass  of 
4  feet  focus  and  37  lines  aperture.  Price  9500  francs. 

12.  Vertical  repeating  circle,  eighteen  inches  in  diameter,  with  an 
azimuth  circle  of  eight  inches.     The  first  is  divided  by  four  verniers 
to  four  seconds  ;  the  latter,  by  one  vernier  to  ten  seconds.     The  tel- 
escope has  an  object-glass  of  2  feet  focus  and  22  lines  aperture. 

Price  3430  francs. 

13.  Vertical  repeating  circle,  fourteen  inches  in  diameter,  with  an 
azimuth  circle  of  six  inches,  divided  like  the  preceding.     The  tele- 
scope has  an  object-glass  of  21  inches  focus  and  22  lines  aperture. 

Price  2580  francs. 

Meridian  Circles  by  Pistor  and  Martins,  Berlin. 

The  following  are  the  prices  of  meridian  circles  and  transit  instru- 
ments made  by  Pistor  and  Martins,  of  Berlin. 

A  rix-dollar  is  equal  to  68  cents  of  United  States  currency. 

1 .  Meridian  circle,  with  a  telescope  of  8  feet  focal  length  and  6 
inches  aperture,  situated  at  the  middle  of  the  horizontal  axis,  with 
four  astronomical  eye-pieces,  magnifying  from  85  to  288  times.     Has 
two  circles  of  four  feet  diameter,  divided  to  single  minutes,  and  each 
furnished  with  four  reading  microscopes.     Price  5000  rix-dollars. 

2.  Meridian  circle,  with  a  telescope  of  6  feet  focus  and  52  lines  ap- 
erture.    Has  two  circles  of  three  feet  diameter,  divided  to  two  min- 
utes, and  reading  by  eight  microscopes.      Price  3800  rix-dollars. 

3.  Meridian  circle,  with  a  telescope  of  4  feet  focus  and  3  inches 


CATALOGUE    OF    INSTRUMENTS.  495 

aperture.     Has  two  circles  of  two  feet  diameter,  divided  to  two  min- 
utes, and  reading  by  eight  microscopes.       Price  2700  rix-dollars. 

4.  Meridian  circle,  with  a  telescope  of  2  feet  focus  and  2  inches 
aperture.     Has  two  circles  of  sixteen  inches  diameter,  reading  with 
four  microscopes.  Price  1000  rix-dollars. 

5.  Transit  instrument,  with  an  8  feet  telescope,  like  No.  1,  omit- 
ting the  circle  and  microscopes.  Price  2400  rix-dollars. 

6.  Transit  instrument,  with  a  6  feet  telescope,  like  No.  2,  without 
the  circle.  Price  1500  rix-dollars. 

7.  Transit  instrument,  with  a  4  feet  telescope,  like  No.  3,  with- 
out the  circle.  Price  700  rix-dollars. 

8.  Transit  instrument,  with  an  8  feet  telescope  at  the  end  of  the 
horizontal  axis.     The  pivots  rest  upon  a  stone  column,  excavated 
in  the  middle,  and  having  a  contrivance  for  rapid  reversal  of  the 
axis,  as  represented  on  page  161.  Price  3800  rix-dollars. 

9.  Transit  instrument,  like  No.  8,  except  a  6  feet  telescope. 

Price  3000  rix-dollars. 

10.  Transit  instrument,  like  No.  8,  except  a  4  feet  telescope. 

Price  2000  rix-dollars. 

11.  Transit  instrument,  like  No.  8,  with  a  three  feet  circle,  read- 
ing by  four  microscopes.  Price  4600  rix-dollars. 

12.  Transit  instrument,  like  No.  9,  with  a  three  feet  circle,  read- 
ing by  four  microscopes.  Price  3800  rix-dollars. 

13.  Transit  instrument,  like  No.  10,  with  a  two  feet  circle,  read- 
ing by  four  microscopes.  Price  2600  rix-dollars. 

Similar  instruments,  and  at  corresponding  prices,  are  made  by  the 
Messrs.  Repsold,  of  Hamburg. 

The  prices  of  the  patent  circles  made  by  Pistor  and  Martins,  Ber- 
lin, and  mentioned  on  page  101,  are  as  follows  : 

Patent  circle,  5  inches  radius,  with  two  verniers,  reading  to  20" 
(or  10",  if  desired) 85  rix-dollars. 

The  same,  with  lamp  for  night  reading 90          " 

Patent  sextant,  6  inches  radius,  reading  to  10".  .    80          " 

Instruments  by  William  Simms,  London. 

1.  Achromatic  telescope,  3f  inches  object-glass  and  five  feet  focal 
length,  mounted  upon  a  universal  equatorial  stand jGHO. 

2.  Achromatic  telescope,  4  inch  object-glass 140. 

3.  Ditto           ditto,        4  inch  object-glass,  mounted  equatorial- 
ly  on  an  iron  pillar,  with  clock  motion j£150. 

4.  Completely  mounted  equatorial,  with  clock  movement,  microm- 
eter, etc. ;  5  feet  focus  and  4  inch  object-glass .£230. 


496  PRACTICAL    ASTRONOMY. 

5.  Equatorial  instruments  of  larger  dimensions,  having  telescopes 
varying  from  4J  to  9  inches  aperture,  with  finely  graduated  circles, 
clock  movement,  micrometers,  etc from  £300  to  800. 

6.  Three  and  a  half  feet  transit  instrument,  constructed  for  fixing 
upon  stone  piers £84. 

7.  Three  and  a  half  feet  transit  instrument  complete,  with  two 
setting  circles,  etc £105. 

8.  Five  feet  transit  instrument,  4  inches  aperture 180. 

9.  Seven  feet  transit  instrument 420. 

10.  Transit  circle,  18  inches,  with  three  reading  microscopes,  hav- 
ing a  telescope  of  30  inches  focus  and  3  inches  aperture  .  .  .£130. 

11 .  Two  feet  transit  circle 220. 

12.  Three  feet  transit  circle 350. 

13.  Four  feet  transit  circle 500. 

14.  Fifteen  inch  altitude  and  azimuth  instrument,  both  circles 
reading  by  micrometers £130. 

15.  Altitude  and  azimuth  instrument,  the  altitude  circle  18  inch- 
es and  azimuth  circle  15  inches,  with  micrometers £150. 

16.  Altitude  and  azimuth  instrument,  both  circles  18  inches,  with 
micrometers £210. 

17.  Twelve  inch  repeating  circle  (Borda's) 84. 

18.  Eighteen  inch          ditto  105. 

19.  Five  or  six  feet  mural  circle 750. 

20.  Eight  feet  mural  circle,  like  that  at  Cambridge  University, 
England £1050. 

A  pound  sterling  is  equal  to  $4.84. 

Telescopes  by  Henry  Fitz,  of  New  York. 

Mr.  Fitz  has  completed  large  telescopes  of  seven  different  sizes. 

No.  1  has  a  clear  aperture  of  12  inches,  and  a  focal  length  of  17 
feet.  It  has  7  negative  and  6  positive  eye-pieces,  the  highest  mag- 
nifying power  being  1200.  The  declination  circle  is  20  inches  in 
diameter,  graduated  to  20X,  and  reads  by  four  verniers  to  20X/.  The 
right  ascension  circle  is  20  inches  in  diameter,  graduated  to  207,  and 
reads  by  two  verniers  to  two  seconds  of  time — is  furnished  with 
clock-work  and  micrometer.  This  telescope  was  sold  to  Michigan 
University  for  $6000. 

No.  2  has  an  aperture  of  9f  inches,  and  a  focal  length  of  14  feet. 
It  has  7  negative  and  6  positive  eye-pieces,  the  highest  magnifying 
power  being  1000.  Circles  of  the  same  size  as  No.  1.  This  tele- 
scope was  sold  to  West  Point  Academy  for  $5000. 

No.  3  has  an  aperture  of  9  inches,  and  a  focal  length  of  9J  feet. 


CATALOGUE    OF    INSTRUMENTS.  497 

Highest  magnifying  power  600.  This  telescope  was  sold  to  Mr. 
Rutherford,  of  New  York,  for  $2200.  It  was  made  with  an  unusu- 
ally short  focus,  to  accommodate  the  size  of  Mr.  Rutherford's  dome. 

No.  4  has  a  focal  length  of  11  feet,  and  an  aperture  of  8J  inches. 
It  has  twelve  eye-pieces,  the  highest  magnifying  800  times.  Price, 
with  clock-work  and  micrometer,  $2200  —  with  plain  mounting, 
$1600. 

No.  5  has  a  focal  length  of  8  feet,  and  an  aperture  of  6J  inches. 
Highest  magnifying  power  500.  Price,  with  clock-work  and  mi- 
crometer, $1300. 

No.  6  has  a  focal  length  of  7  feet,  and  an  aperture  of  5  inches. 
Highest  magnifying  power  400  times.  Price,  with  clock-work  and 
micrometer,  $1050 — without  clock-work  or  micrometer,  $825. 

No.  7  has  a  focal  length  of  5  feet,  and  an  aperture  of  4  inches. 
Highest  magnifying  power  250  times.  Price  $225,  without  clock- 
work or  micrometer. 

Mr.  Fitz  obtains  his  crown  glass  from  the  manufactory  of  Bon- 
temps,  of  Birmingham,  England  ;  his  flint  glass  he  obtains  from  Paris. 

Several  of  Mr.  Fitz's  telescopes  have  been  subjected  to  the  se- 
verest tests  by  competent  judges,  and  have  been  decided  to  com- 
pare favorably  with  the  best  Munich  instruments. 

Telescopes  by  Alvan  Clark,  of  Boston. 

Mr.  Clark  has  ground  and  mounted  twelve  object-glasses  of  from 
4  to  7ij  inches  aperture,  the  largest  of  which  was  purchased  by  an 
eminent  English  observer.  Two  fine  double  stars  were  discovered 
with  this  glass  by  Mr.  Clark,  one  of  which  is  25  Ceti.  Mr.  Clark  has 
just  completed  an  instrument  for  Amherst  College,  7J-  inches  aper- 
ture, and  a  focal  distance  of  101  inches,  having  a  pendulum-driv- 
ing clock,  to  which  is  applied  Mr.  Bond's  spring  governor.  The  tel- 
escope was  furnished  at  a  cost  of  $1800.  It  has  brought  to  view 
two  new  double  stars,  one  in  R.  A.,  18h.  17m.  11s. ;  Dec.,  1°  39X  23" 
S.;  Mag.,  7J  and  7J ;  Distance  estimated,  0".3  :  the  other  is  in  R. 
A.,  19h.  50m.  35s. ;  Dec.,  2°  38/  1"  S. ;  Mag.,  7£  and  8  ;  Distance, 
0".8. 

Il 


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